Multiple Correct Answers MCQs for Sub-Topics of Topic 12: Vectors & Three-Dimensional Geometry

Question 1. Which of the following quantities are vectors?

(A) Displacement

(B) Electric field intensity

(C) Volume

(D) Momentum

Question 2. If a vector $\vec{v}$ has magnitude $M$ and direction given by angle $\theta$ with the positive x-axis in a 2D plane, which of the following are correct representations of $\vec{v}$?

(A) $(M \cos \theta, M \sin \theta)$

(B) $M \cos \theta \hat{i} + M \sin \theta \hat{j}$

(C) $M \hat{u}$, where $\hat{u}$ is a unit vector in the direction of $\vec{v}$.

(D) $(M, \theta)$

Question 3. Which of the following statements are true about types of vectors?

(A) A unit vector has magnitude zero.

(B) Equal vectors must have the same magnitude and direction.

(C) Coinitial vectors have the same initial point.

(D) Collinear vectors lie on the same line or parallel lines.

Question 4. Let $\vec{a} = 3\hat{i} - \hat{j}$ and $\vec{b} = -\hat{i} + 2\hat{j}$. Which of the following are true?

(A) $\vec{a} + \vec{b} = 2\hat{i} + \hat{j}$

(B) $\vec{a} - \vec{b} = 4\hat{i} - 3\hat{j}$

(C) $2\vec{a} = 6\hat{i} - 2\hat{j}$

(D) $|\vec{a}| = \sqrt{10}$

Question 5. For vectors $\vec{u}, \vec{v}, \vec{w}$ and scalars $c_1, c_2$, which of the following are properties of vector algebra?

(A) $\vec{u} + \vec{v} = \vec{v} + \vec{u}$

(B) $(\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w})$

(C) $c_1(c_2 \vec{u}) = (c_1 c_2) \vec{u}$

(D) $c_1(\vec{u} + \vec{v}) = c_1 \vec{u} + c_1 \vec{v}$

Question 6. Consider the vectors $\vec{p} = \hat{i} + 2\hat{j}$ and $\vec{q} = 2\hat{i} + 4\hat{j}$. Which of the following are true?

(A) $\vec{p}$ and $\vec{q}$ are collinear.

(B) $\vec{q} = 2\vec{p}$.

(C) $|\vec{p}| = \sqrt{5}$.

(D) The angle between $\vec{p}$ and $\vec{q}$ is $0^\circ$.

Question 7. If $\vec{a}$ is a non-zero vector, then a unit vector in the direction opposite to $\vec{a}$ is:

(A) $-\frac{\vec{a}}{|\vec{a}|}$

(B) $\frac{-\vec{a}}{|\vec{a}|}$

(C) A vector with magnitude 1.

(D) A vector collinear with $\vec{a}$.

Question 8. Let $\vec{v}$ be a vector. Which of the following statements are always true?

(A) $\vec{v} + (-\vec{v}) = \vec{0}$

(B) $\vec{v} + \vec{0} = \vec{v}$

(C) $|\vec{v}| \geq 0$

(D) $|\vec{v} + \vec{w}| = |\vec{v}| + |\vec{w}|$

Question 9. Consider two vectors $\vec{a}$ and $\vec{b}$ represented by the adjacent sides of a triangle. Which of the following represents the third side?

(A) $\vec{a} + \vec{b}$ (if sides are taken in order)

(B) $\vec{b} - \vec{a}$

(C) $\vec{a} - \vec{b}$

(D) $\vec{a} + \vec{b}$ (if sides are taken as initial point to terminal point forming the triangle)

Question 10. Which of the following statements are correct regarding scalar multiplication by 0?

(A) $0 \cdot \vec{a} = \vec{0}$ for any vector $\vec{a}$.

(B) $0 \cdot k = 0$ for any scalar $k$.

(C) If $k\vec{a} = \vec{0}$, then either $k=0$ or $\vec{a} = \vec{0}$ (or both).

(D) The magnitude of $0 \cdot \vec{a}$ is always 0.

Question 11. If $\vec{a}$ and $\vec{b}$ are two vectors, then $|\vec{a} + \vec{b}|^2$ is equal to:

(A) $(\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b})$

(B) $|\vec{a}|^2 + |\vec{b}|^2 + 2\vec{a} \cdot \vec{b}$

(C) $|\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}|\cos\theta$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.

(D) $|\vec{a}|^2 + |\vec{b}|^2$

Question 1. Let A be the point $(1, 2, -3)$ and B be the point $(3, -1, 4)$. Which of the following statements are true?

(A) The position vector of A is $\hat{i} + 2\hat{j} - 3\hat{k}$.

(B) The vector $\vec{AB} = 2\hat{i} - 3\hat{j} + 7\hat{k}$.

(C) The magnitude of $\vec{AB}$ is $\sqrt{4+9+49} = \sqrt{62}$.

(D) The vector $\vec{BA} = (1-3)\hat{i} + (2-(-1))\hat{j} + (-3-4)\hat{k} = -2\hat{i} + 3\hat{j} - 7\hat{k}$.

Question 2. Consider the vectors $\vec{a} = \hat{i} - \hat{j}$ and $\vec{b} = -2\hat{i} + 2\hat{j}$. Which of the following are true?

(A) $\vec{a}$ and $\vec{b}$ are collinear.

(B) $\vec{b} = -2\vec{a}$.

(C) $\vec{a} + \vec{b} = -\hat{i} + \hat{j}$.

(D) $|\vec{b}| = 2\sqrt{(-1)^2 + (1)^2} = 2\sqrt{2}$.

Question 3. Let $\vec{u} = \hat{i} + 2\hat{j}$ and $\vec{v} = 3\hat{i} - \hat{j}$. Which of the following vectors can be expressed as a linear combination of $\vec{u}$ and $\vec{v}$?

(A) $4\hat{i} + \hat{j}$

(B) $5\hat{i} + 3\hat{j}$

(C) $\hat{k}$

(D) $\vec{0}$

Question 4. Three vectors $\vec{a}, \vec{b}, \vec{c}$ with non-zero magnitudes are coplanar. Which of the following conditions can imply coplanarity?

(A) $[\vec{a}, \vec{b}, \vec{c}] = 0$

(B) One vector is a linear combination of the other two (assuming the other two are non-collinear).

(C) $\vec{a} \cdot (\vec{b} \times \vec{c}) = 0$

(D) There exist scalars $x, y, z$, not all zero, such that $x\vec{a} + y\vec{b} + z\vec{c} = \vec{0}$.

Question 5. The components of the vector joining point P(2,-1,3) to Q(1,2,-1) are:

(A) $(1-2, 2-(-1), -1-3)$

(B) $(-1, 3, -4)$

(C) $1\hat{i} + 3\hat{j} - 4\hat{k}$

(D) $\vec{QP}$ components $(1, -3, 4)$

Question 6. In 3D space, any vector can be uniquely expressed as a linear combination of:

(A) Any three non-collinear vectors.

(B) Any three non-coplanar vectors.

(C) The standard basis vectors $\hat{i}, \hat{j}, \hat{k}$.

(D) Any vector and its scalar multiple.

Question 7. If the position vector of a point P is $2\hat{i} - \hat{j} + 3\hat{k}$, then:

(A) The coordinates of P are $(2, -1, 3)$.

(B) The distance of P from the origin is $\sqrt{4+1+9} = \sqrt{14}$.

(C) The vector from the origin to P is $2\hat{i} - \hat{j} + 3\hat{k}$.

(D) The unit vector in the direction of OP is $\frac{1}{\sqrt{14}}(2\hat{i} - \hat{j} + 3\hat{k})$.

Question 8. Which of the following sets of vectors are collinear?

(A) $\{\hat{i} + \hat{j}, 2\hat{i} + 2\hat{j}\}$

(B) $\{\hat{i} - \hat{j}, \hat{j} - \hat{k}, \hat{k} - \hat{i}\}$

(C) $\{3\hat{i} - 2\hat{j} + \hat{k}, -6\hat{i} + 4\hat{j} - 2\hat{k}\}$

(D) $\{\hat{i}, \hat{j}\}$

Question 9. If vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar, which of the following is true?

(A) They lie in the same plane.

(B) One vector can be expressed as a linear combination of the other two (if they are non-collinear).

(C) Their scalar triple product is zero.

(D) They are linearly dependent.

Question 10. The theorem on two non-zero non-collinear vectors in 2D states that they form a basis. This implies that:

(A) Any vector in the 2D plane can be written as a linear combination of these two vectors.

(B) The linear combination is unique.

(C) The two vectors are linearly independent.

(D) The two vectors span the entire 2D plane.

Question 11. The magnitude of the vector $\vec{v} = x\hat{i} + y\hat{j} + z\hat{k}$ is:

(A) $\sqrt{x^2 + y^2 + z^2}$

(B) $|\vec{v}|$

(C) $\sqrt{(x\hat{i})^2 + (y\hat{j})^2 + (z\hat{k})^2}$

(D) The distance of point $(x,y,z)$ from origin $(0,0,0)$.

Question 1. Which of the following are properties of the scalar (dot) product?

(A) $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$ (Commutativity)

(B) $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$ (Distributivity)

(C) $(k\vec{a}) \cdot \vec{b} = k(\vec{a} \cdot \vec{b})$ for scalar $k$.

(D) $\vec{a} \cdot \vec{a} = |\vec{a}|^2$

Question 2. If $\vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$, then $\vec{a} \cdot \vec{b}$ is:

(A) $2(1) + (-1)(2) + 3(-1) = 2 - 2 - 3 = -3$

(B) A scalar value.

(C) Equal to $\vec{b} \cdot \vec{a}$.

(D) A vector quantity.

Question 3. If $\vec{a}$ and $\vec{b}$ are two non-zero vectors, $\vec{a} \cdot \vec{b} = 0$ implies:

(A) The angle between $\vec{a}$ and $\vec{b}$ is $90^\circ$ or $\pi/2$ radians.

(B) $\vec{a}$ is perpendicular to $\vec{b}$.

(C) The projection of $\vec{a}$ on $\vec{b}$ is zero.

(D) The projection of $\vec{b}$ on $\vec{a}$ is zero.

Question 4. The projection of vector $\vec{a}$ on vector $\vec{b}$ can be represented by:

(A) $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$ (scalar projection)

(B) $(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}) \vec{b}$ (vector projection)

(C) $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|}$ (scalar projection of $\vec{b}$ on $\vec{a}$)

(D) A scalar quantity.

Question 5. The work done by a constant force $\vec{F}$ when its point of application is displaced by $\vec{d}$ is given by $\vec{W} = \vec{F} \cdot \vec{d}$. This implies:

(A) Work done is a scalar quantity.

(B) If $\vec{F}$ is perpendicular to $\vec{d}$, the work done is zero.

(C) If the angle between $\vec{F}$ and $\vec{d}$ is acute, work done is positive.

(D) If the angle between $\vec{F}$ and $\vec{d}$ is obtuse, work done is negative.

Question 6. For any two vectors $\vec{a}$ and $\vec{b}$, the following statements are true:

(A) $\vec{a} \cdot \vec{a} \geq 0$

(B) $(\vec{a} + \vec{b})^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2\vec{a} \cdot \vec{b}$

(C) $(\vec{a} - \vec{b})^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2\vec{a} \cdot \vec{b}$

(D) $|\vec{a} \cdot \vec{b}| \leq |\vec{a}||\vec{b}|$ (Cauchy-Schwarz inequality)

Question 7. If $\vec{a}$ and $\vec{b}$ are non-zero vectors, the angle $\theta$ between them can be calculated using the dot product:

(A) $\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$

(B) $\theta = \cos^{-1}\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\right)$

(C) If $\vec{a} \cdot \vec{b} > 0$, $\theta$ is acute.

(D) If $\vec{a} \cdot \vec{b} < 0$, $\theta$ is obtuse.

Question 8. The dot product is used in physics to calculate:

(A) Work done by a force.

(B) Power.

(C) The component of a force in a given direction.

(D) Torque.

Question 9. The scalar product in terms of components is given by $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$ for $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$. Using this, which of the following are true?

(A) $\hat{i} \cdot \hat{i} = 1$

(B) $\hat{i} \cdot \hat{j} = 0$

(C) $\hat{i} \cdot \hat{k} = 0$

(D) $\hat{i} \cdot (\hat{j} + \hat{k}) = 0$

Question 10. If $|\vec{a}| = 5$ and $\vec{a} \cdot \vec{b} = 10$, the scalar projection of $\vec{b}$ on $\vec{a}$ is:

(A) $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|} = \frac{10}{5} = 2$

(B) 2

(C) A scalar quantity.

(D) A vector quantity.

Question 11. For any vector $\vec{v}$, $\vec{v} \cdot \vec{v}$ equals:

(A) $|\vec{v}|^2$

(B) A non-negative scalar.

(C) $v_x^2 + v_y^2 + v_z^2$, where $v_x, v_y, v_z$ are components of $\vec{v}$.

(D) $\vec{v}$

Question 1. Which of the following are properties of the vector (cross) product?

(A) $\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$ (Anti-commutativity)

(B) $(\vec{a} + \vec{b}) \times \vec{c} = \vec{a} \times \vec{c} + \vec{b} \times \vec{c}$ (Distributivity)

(C) $(k\vec{a}) \times \vec{b} = k(\vec{a} \times \vec{b})$ for scalar $k$.

(D) $\vec{a} \times \vec{a} = \vec{0}$

Question 2. If $\vec{a} = \hat{i} + \hat{j}$ and $\vec{b} = \hat{k}$, find $\vec{a} \times \vec{b}$.

(A) $\hat{i} \times \hat{k} + \hat{j} \times \hat{k}$

(B) $-\hat{j} + \hat{i}$

(C) $\hat{i} - \hat{j}$

(D) A vector perpendicular to both $\hat{i} + \hat{j}$ and $\hat{k}$.

Question 3. If $\vec{a}$ and $\vec{b}$ are two non-zero vectors, $\vec{a} \times \vec{b} = \vec{0}$ implies:

(A) The angle between $\vec{a}$ and $\vec{b}$ is $0^\circ$ or $180^\circ$ (0 or $\pi$ radians).

(B) $\vec{a}$ is parallel to $\vec{b}$.

(C) $\vec{a}$ is collinear with $\vec{b}$.

(D) The magnitude $|\vec{a}||\vec{b}|\sin\theta = 0$.

Question 4. The magnitude of the cross product $|\vec{a} \times \vec{b}|$ represents:

(A) The area of the parallelogram formed by adjacent sides $\vec{a}$ and $\vec{b}$.

(B) Twice the area of the triangle formed by adjacent sides $\vec{a}$ and $\vec{b}$.

(C) $|\vec{a}||\vec{b}|\sin\theta$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.

(D) A scalar quantity.

Question 5. The direction of the vector $\vec{a} \times \vec{b}$ is:

(A) Perpendicular to the plane containing $\vec{a}$ and $\vec{b}$.

(B) Given by the right-hand rule.

(C) A unit vector $\hat{n}$ such that $\vec{a}, \vec{b}, \hat{n}$ form a right-handed system.

(D) Always along the z-axis.

Question 6. If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, the cross product $\vec{a} \times \vec{b}$ can be calculated using the determinant:

(A) $\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

(B) $(a_2b_3 - a_3b_2)\hat{i} - (a_1b_3 - a_3b_1)\hat{j} + (a_1b_2 - a_2b_1)\hat{k}$

(C) A vector quantity.

(D) $\vec{a} \cdot \vec{b}$

Question 7. Applications of the cross product include finding:

(A) The area of a parallelogram.

(B) The area of a triangle.

(C) A normal vector to a plane defined by two vectors.

(D) Torque produced by a force.

Question 8. For orthonormal basis vectors $\hat{i}, \hat{j}, \hat{k}$, which of the following are true?

(A) $\hat{i} \times \hat{j} = \hat{k}$

(B) $\hat{j} \times \hat{k} = \hat{i}$

(C) $\hat{k} \times \hat{i} = \hat{j}$

(D) $\hat{j} \times \hat{i} = \hat{k}$

Question 9. If $\vec{a}$ and $\vec{b}$ are vectors such that $|\vec{a}|=3$, $|\vec{b}|=4$, and the angle between them is $30^\circ$, then $|\vec{a} \times \vec{b}|$ is:

(A) $|\vec{a}||\vec{b}|\sin 30^\circ = 3 \times 4 \times \frac{1}{2} = 6$

(B) 6

(C) A non-negative scalar value.

(D) $12 \times \frac{1}{2}$

Question 10. Let $\vec{u}$ and $\vec{v}$ be vectors. If $\vec{u} \times \vec{v} = \vec{w}$, then:

(A) $\vec{w}$ is perpendicular to $\vec{u}$.

(B) $\vec{w}$ is perpendicular to $\vec{v}$.

(C) $\vec{u} \cdot \vec{w} = 0$.

(D) $\vec{v} \cdot \vec{w} = 0$.

Question 11. The area of the triangle with vertices A(1,0,0), B(0,1,0), C(0,0,1) can be found by considering vectors $\vec{AB}$ and $\vec{AC}$ as adjacent sides. Which calculation steps are correct?

(A) $\vec{AB} = -\hat{i} + \hat{j}$

(B) $\vec{AC} = -\hat{i} + \hat{k}$

(C) $\vec{AB} \times \vec{AC} = (\hat{i} + \hat{j} + \hat{k})$

(D) Area is $\frac{1}{2} |\vec{AB} \times \vec{AC}| = \frac{1}{2} \sqrt{1^2+1^2+1^2} = \frac{\sqrt{3}}{2}$.

Question 1. The scalar triple product $[\vec{a}, \vec{b}, \vec{c}]$ is defined as $\vec{a} \cdot (\vec{b} \times \vec{c})$. Which of the following are true about this product?

(A) It is a scalar quantity.

(B) It can be calculated using a determinant of the vector components.

(C) Its value is independent of the order of dot and cross products, i.e., $\vec{a} \cdot (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}) \cdot \vec{c}$.

(D) The value changes sign if the vectors are cyclically permuted.

Question 2. The geometrical interpretation of the scalar triple product $|[\vec{a}, \vec{b}, \vec{c}]|$ is:

(A) The area of the parallelogram formed by $\vec{a}$ and $\vec{b}$.

(B) The volume of the parallelepiped with adjacent edges $\vec{a}, \vec{b}, \vec{c}$.

(C) Six times the volume of the tetrahedron with adjacent edges $\vec{a}, \vec{b}, \vec{c}$.

(D) A non-negative real number representing volume.

Question 3. Three vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar if and only if:

(A) $[\vec{a}, \vec{b}, \vec{c}] = 0$.

(B) $\vec{a} \cdot (\vec{b} \times \vec{c}) = 0$.

(C) One vector is a linear combination of the other two (provided the other two are non-collinear).

(D) The volume of the parallelepiped formed by them is zero.

Question 4. If two vectors in a scalar triple product are identical, e.g., $[\vec{a}, \vec{a}, \vec{b}]$, the value is:

(A) 0

(B) $\vec{a} \cdot (\vec{a} \times \vec{b})$

(C) A scalar.

(D) Non-zero unless $\vec{a} = \vec{0}$.

Question 5. Consider vectors $\vec{a} = \hat{i}$, $\vec{b} = \hat{j}$, $\vec{c} = \hat{k}$. Which of the following are true?

(A) $[\vec{a}, \vec{b}, \vec{c}] = 1$

(B) $[\hat{i}, \hat{j}, \hat{k}] = 1$

(C) These vectors are non-coplanar.

(D) They form a right-handed system.

Question 6. For vectors $\vec{a}, \vec{b}, \vec{c}$, which of the following scalar triple products have the same value as $[\vec{a}, \vec{b}, \vec{c}]$?

(A) $[\vec{b}, \vec{c}, \vec{a}]$

(B) $[\vec{c}, \vec{a}, \vec{b}]$

(C) $[\vec{a}, \vec{c}, \vec{b}]$

(D) $\vec{c} \cdot (\vec{a} \times \vec{b})$

Question 7. If the scalar triple product $[\vec{u}, \vec{v}, \vec{w}] \neq 0$, then which of the following must be true?

(A) $\vec{u}, \vec{v}, \vec{w}$ are linearly independent.

(B) $\vec{u}, \vec{v}, \vec{w}$ are non-coplanar.

(C) They can form a basis in 3D space.

(D) None of the vectors is the zero vector.

Question 8. The volume of the tetrahedron with vertices P, Q, R, S is given by $\frac{1}{6} |[\vec{PQ}, \vec{PR}, \vec{PS}]|$. This formula is valid if:

(A) P, Q, R, S are non-coplanar.

(B) P, Q, R are non-collinear.

(C) $[\vec{PQ}, \vec{PR}, \vec{PS}] \neq 0$.

(D) The points P, Q, R, S form a tetrahedron.

Question 9. Consider the vectors $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$, $\vec{c} = -\hat{i} + 2\hat{j}$. Which of the following are true?

(A) $[\vec{a}, \vec{b}, \vec{c}] = \begin{vmatrix} 1 & 1 & 1 \\ 2 & -1 & 1 \\ -1 & 2 & 0 \end{vmatrix}$

(B) $[\vec{a}, \vec{b}, \vec{c}] = 1(0-2) - 1(0-(-1)) + 1(4-1) = -2 - 1 + 3 = 0$.

(C) The vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar.

(D) The volume of the parallelepiped formed by these vectors is 0.

Question 10. If $[\vec{a}, \vec{b}, \vec{c}] = 5$, then $[\vec{a}, \vec{c}, \vec{b}]$ is:

(A) $-5$

(B) The negative of $[\vec{a}, \vec{b}, \vec{c}]$.

(C) $\vec{a} \cdot (\vec{c} \times \vec{b})$

(D) A scalar quantity.

Question 11. Which of the following is always equal to $\vec{a} \cdot (\vec{b} \times \vec{c})$?

(A) $(\vec{a} \times \vec{b}) \cdot \vec{c}$

(B) $[\vec{a}, \vec{b}, \vec{c}]$

(C) $\vec{b} \cdot (\vec{c} \times \vec{a})$

(D) $\vec{c} \cdot (\vec{a} \times \vec{b})$

Question 12. If the volume of the parallelepiped formed by $\vec{u}, \vec{v}, \vec{w}$ is $V$, then the volume of the parallelepiped formed by $2\vec{u}, 3\vec{v}, \vec{w}$ is:

(A) $6V$

(B) $|[2\vec{u}, 3\vec{v}, \vec{w}]| = |2 \times 3 \times 1 \times [\vec{u}, \vec{v}, \vec{w}]| = 6|[\vec{u}, \vec{v}, \vec{w}]| = 6V$

(C) A scalar value.

(D) Dependent on the scalars multiplying the vectors.

Question 1. The position vector of a point R dividing the line segment joining points A ($\vec{a}$) and B ($\vec{b}$) internally in the ratio $m:n$ is $\vec{r} = \frac{n\vec{a} + m\vec{b}}{m+n}$. Which of the following are true?

(A) If R is the midpoint, $m=n=1$, so $\vec{r} = \frac{\vec{a} + \vec{b}}{2}$.

(B) If R is the midpoint, the ratio is $1:1$.

(C) The formula can be derived using collinearity of A, R, B and vector addition.

(D) If $m=1, n=0$, R coincides with B.

Question 2. The position vector of a point R dividing the line segment joining points A ($\vec{a}$) and B ($\vec{b}$) externally in the ratio $m:n$ is $\vec{r} = \frac{m\vec{b} - n\vec{a}}{m-n}$. Which of the following are true?

(A) $m \neq n$ for external division.

(B) If $m>n$, R lies on the extension of AB beyond B.

(C) If $m

(D) If $m=2, n=1$, $\vec{r} = \frac{2\vec{b} - \vec{a}}{1} = 2\vec{b} - \vec{a}$.

Question 3. Let the vertices of a triangle be A, B, C with position vectors $\vec{a}, \vec{b}, \vec{c}$. The centroid G of the triangle has position vector $\vec{g} = \frac{\vec{a} + \vec{b} + \vec{c}}{3}$. Which of the following are true?

(A) The centroid divides the median from A to the midpoint of BC (say M) in the ratio 2:1.

(B) The position vector of M is $\frac{\vec{b}+\vec{c}}{2}$.

(C) $\vec{g} = \frac{1 \cdot \vec{a} + 2 \cdot \vec{m}}{1+2} = \frac{\vec{a} + 2(\frac{\vec{b}+\vec{c}}{2})}{3} = \frac{\vec{a} + \vec{b} + \vec{c}}{3}$.

(D) $\vec{GA} + \vec{GB} + \vec{GC} = \vec{0}$.

Question 4. If the position vectors of P and Q are $\vec{p}$ and $\vec{q}$, and R is a point on the line PQ such that $\vec{OR} = \frac{5\vec{q} - 2\vec{p}}{3}$, then:

(A) R divides PQ externally in the ratio 5:2.

(B) R divides QP externally in the ratio 2:5.

(C) R lies outside the line segment PQ.

(D) $\vec{PR} = \frac{5}{3} \vec{PQ}$.

Question 5. The position vectors of the vertices of a tetrahedron A, B, C, D are $\vec{a}, \vec{b}, \vec{c}, \vec{d}$. The centroid G of the tetrahedron is the point that divides the line segment joining any vertex to the centroid of the opposite face in the ratio 3:1. Which of the following are true?

(A) The centroid of face BCD has position vector $\frac{\vec{b}+\vec{c}+\vec{d}}{3}$.

(B) The position vector of G is $\frac{1 \cdot \vec{a} + 3 \cdot (\frac{\vec{b}+\vec{c}+\vec{d}}{3})}{1+3} = \frac{\vec{a} + \vec{b} + \vec{c} + \vec{d}}{4}$.

(C) The position vector of G is the average of the position vectors of the four vertices.

(D) $\vec{GA} + \vec{GB} + \vec{GC} + \vec{GD} = \vec{0}$.

Question 6. If point C divides line segment AB such that $\vec{AC} = k \vec{CB}$ for some scalar $k$. Which of the following are true?

(A) If $k>0$, C divides AB internally.

(B) If $k<0$, C divides AB externally.

(C) If $k=1$, C is the midpoint of AB.

(D) If $k = m/n$, C divides AB in the ratio $m:n$.

Question 7. Let A=(1,2) and B=(4,8). Find the coordinates of point C that divides AB internally in the ratio 1:2.

(A) The position vector of C is $\frac{2(1\hat{i} + 2\hat{j}) + 1(4\hat{i} + 8\hat{j})}{1+2}$

(B) The position vector of C is $\frac{2\hat{i} + 4\hat{j} + 4\hat{i} + 8\hat{j}}{3} = \frac{6\hat{i} + 12\hat{j}}{3} = 2\hat{i} + 4\hat{j}$.

(C) The coordinates of C are (2,4).

(D) Point C lies between A and B.

Question 8. The section formula in vector form is a powerful tool for dealing with collinear points. Which of the following can be derived from the section formula?

(A) Midpoint formula.

(B) Condition for collinearity of three points.

(C) Position vector of the centroid of a triangle.

(D) Equation of a line passing through two points.

Question 9. Let $\vec{a}$ and $\vec{b}$ be position vectors of points A and B. If R is a point such that $3\vec{OR} = 2\vec{OA} + \vec{OB}$ (where O is the origin), then:

(A) $\vec{r} = \frac{2\vec{a} + \vec{b}}{3}$.

(B) R divides AB internally in the ratio 1:2.

(C) R lies on the line segment AB.

(D) R divides BA internally in the ratio 2:1.

Question 10. If the position vectors of A, B, C are $\vec{a}, \vec{b}, \vec{c}$ respectively, and $2\vec{a} + 3\vec{b} - 5\vec{c} = \vec{0}$, then which of the following are true?

(A) A, B, C are collinear.

(B) C divides AB internally in the ratio 3:2.

(C) $\vec{c} = \frac{2\vec{a} + 3\vec{b}}{5}$.

(D) A, B, C form a triangle.

Question 11. The position vector of a point P dividing the line segment joining A(2,1,4) and B(4,3,2) externally in the ratio 1:2 is:

(A) $\frac{1(4\hat{i} + 3\hat{j} + 2\hat{k}) - 2(2\hat{i} + \hat{j} + 4\hat{k})}{1-2}$

(B) $\frac{4\hat{i} + 3\hat{j} + 2\hat{k} - 4\hat{i} - 2\hat{j} - 8\hat{k}}{-1} = \frac{\hat{j} - 6\hat{k}}{-1} = -\hat{j} + 6\hat{k}$.

(C) Point P has coordinates $(0, -1, 6)$.

(D) P divides BA externally in the ratio 2:1.

Question 1. In a 3D coordinate system, which of the following statements are true?

(A) The equation $x=0$ represents the yz-plane.

(B) The equation $x=a$ represents a plane parallel to the yz-plane.

(C) The equation $z=0$ represents the xy-plane.

(D) The origin is the point $(0,0,0)$.

Question 2. The distance between two points P$(x_1, y_1, z_1)$ and Q$(x_2, y_2, z_2)$ in 3D space is given by the distance formula. Which of the following are correct expressions for the distance PQ?

(A) $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$

(B) $|\vec{PQ}|$, where $\vec{PQ} = (x_2-x_1)\hat{i} + (y_2-y_1)\hat{j} + (z_2-z_1)\hat{k}$.

(C) $\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2}$

(D) $|\vec{OP} - \vec{OQ}|$, where O is the origin.

Question 3. If a line has direction ratios $a, b, c$, which of the following can be its direction cosines?

(A) $(\frac{a}{\sqrt{a^2+b^2+c^2}}, \frac{b}{\sqrt{a^2+b^2+c^2}}, \frac{c}{\sqrt{a^2+b^2+c^2}})$

(B) $(ka, kb, kc)$ for any scalar $k \neq 0$.

(C) $(\pm \frac{a}{\sqrt{a^2+b^2+c^2}}, \pm \frac{b}{\sqrt{a^2+b^2+c^2}}, \pm \frac{c}{\sqrt{a^2+b^2+c^2}})$ (with consistent signs)

(D) $(\ell, m, n)$ such that $\ell^2 + m^2 + n^2 = 1$.

Question 4. The vector equation of a line passing through point A with position vector $\vec{a}$ and parallel to vector $\vec{b}$ is $\vec{r} = \vec{a} + \lambda \vec{b}$. Which of the following are true about this equation?

(A) $\vec{a}$ is the position vector of a point on the line.

(B) $\vec{b}$ is the direction vector of the line.

(C) $\lambda$ is a scalar parameter that can take any real value.

(D) For $\lambda=0$, $\vec{r} = \vec{a}$.

Question 5. The Cartesian equation of a line is $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$. Which of the following are true?

(A) The line passes through the point $(x_1, y_1, z_1)$.

(B) The direction ratios of the line are proportional to $(a, b, c)$.

(C) The vector parallel to the line is $a\hat{i} + b\hat{j} + c\hat{k}$.

(D) The equation is valid even if $a, b,$ or $c$ are zero (interpreting $\frac{x-x_1}{0}$ as $x-x_1=0$).

Question 6. The vector equation of the line passing through points A ($\vec{a}$) and B ($\vec{b}$) is $\vec{r} = (1-\lambda)\vec{a} + \lambda\vec{b}$. Which of the following are true?

(A) For $\lambda=0$, $\vec{r} = \vec{a}$, which is point A.

(B) For $\lambda=1$, $\vec{r} = \vec{b}$, which is point B.

(C) The direction vector of the line is $\vec{b} - \vec{a}$.

(D) For $0 < \lambda < 1$, the point $\vec{r}$ lies on the line segment AB.

Question 7. The line $\frac{x-2}{3} = \frac{y+1}{4} = \frac{z-5}{2}$ passes through which of the following points?

(A) $(2, -1, 5)$

(B) $(5, 3, 7)$ (by setting the ratio to 1)

(C) $(-1, -5, 3)$ (by setting the ratio to -1)

(D) $(8, 7, 9)$ (by setting the ratio to 2)

Question 8. If the direction cosines of a line are $(\ell, m, n)$, then:

(A) $\ell^2 + m^2 + n^2 = 1$.

(B) $(\ell, m, n)$ represent the cosine of the angles the line makes with the positive x, y, and z axes respectively.

(C) Direction ratios are proportional to $(\ell, m, n)$.

(D) There are two possible sets of direction cosines for any given line, $(\ell, m, n)$ and $(-\ell, -m, -n)$.

Question 9. The position vector of a point P on the line $\vec{r} = (2\hat{i} - \hat{j} + \hat{k}) + \lambda (\hat{i} + 3\hat{j} - 2\hat{k})$ can be written as:

(A) $(2+\lambda)\hat{i} + (-1+3\lambda)\hat{j} + (1-2\lambda)\hat{k}$

(B) $(2+\lambda, -1+3\lambda, 1-2\lambda)$ in coordinate form.

(C) $\vec{OP}$ where O is the origin.

(D) A linear combination of $(2\hat{i} - \hat{j} + \hat{k})$ and $(\hat{i} + 3\hat{j} - 2\hat{k})$.

Question 10. The section formula in 3D is similar to the 2D case. If A and B are points with position vectors $\vec{a}$ and $\vec{b}$, and R divides AB internally in ratio $m:n$, then the coordinates of R $(x,y,z)$ are given by:

(A) $x = \frac{nx_A + mx_B}{m+n}$

(B) $y = \frac{ny_A + my_B}{m+n}$

(C) $z = \frac{nz_A + mz_B}{m+n}$

(D) $\vec{r} = \frac{n\vec{a} + m\vec{b}}{m+n}$

Question 11. Which of the following represent the same line?

(A) $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda (2\hat{i} + 4\hat{j} + 6\hat{k})$

(B) $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \mu (\hat{i} + 2\hat{j} + 3\hat{k})$

(C) $\frac{x-1}{2} = \frac{y-2}{4} = \frac{z-3}{6}$

(D) $\frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3}$

Question 1. The equation of a plane can be represented in various forms. Which of the following are valid representations?

(A) Vector form: $\vec{r} \cdot \vec{n} = d$

(B) Cartesian form: $Ax + By + Cz + D = 0$

(C) Intercept form: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$

(D) Vector form (Point-Normal): $(\vec{r} - \vec{a}) \cdot \vec{n} = 0$

Question 2. The Cartesian equation of the plane $\vec{r} \cdot (3\hat{i} - 2\hat{j} + 5\hat{k}) = 7$ is:

(A) $3x - 2y + 5z = 7$

(B) $3x - 2y + 5z - 7 = 0$

(C) A linear equation in x, y, and z.

(D) $3x\hat{i} - 2y\hat{j} + 5z\hat{k} = 7$

Question 3. The equation $2x - 3y + 4z = 12$ represents a plane. Which of the following are true?

(A) The normal vector to the plane is $2\hat{i} - 3\hat{j} + 4\hat{k}$.

(B) The intercepts on the axes are 6, -4, and 3.

(C) The distance from the origin is $\frac{|-12|}{\sqrt{2^2+(-3)^2+4^2}} = \frac{12}{\sqrt{4+9+16}} = \frac{12}{\sqrt{29}}$.

(D) The intercept form is $\frac{x}{6} + \frac{y}{-4} + \frac{z}{3} = 1$.

Question 4. The equation of any plane passing through the origin $(0,0,0)$ must be of the form $Ax + By + Cz = 0$. Which of the following planes pass through the origin?

(A) $x+y+z=0$

(B) $\vec{r} \cdot (\hat{i} - \hat{j} + 2\hat{k}) = 0$

(C) The plane perpendicular to $2\hat{i} + \hat{j} - \hat{k}$ at the origin.

(D) The plane with intercept form $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$.

Question 5. To find the equation of a plane passing through three non-collinear points A, B, C, one method is to find two non-collinear vectors in the plane, e.g., $\vec{AB}$ and $\vec{AC}$. Then a normal vector to the plane is given by their cross product. Which of the following are correct steps or statements?

(A) Calculate $\vec{AB}$ and $\vec{AC}$.

(B) Calculate $\vec{n} = \vec{AB} \times \vec{AC}$.

(C) The equation of the plane is $(\vec{r} - \vec{a}) \cdot \vec{n} = 0$, where $\vec{a}$ is the position vector of A.

(D) The equation can also be written as $\vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n}$.

Question 6. The equation of any plane passing through the intersection of planes $P_1: A_1x + B_1y + C_1z + D_1 = 0$ and $P_2: A_2x + B_2y + C_2z + D_2 = 0$ is given by $P_1 + \lambda P_2 = 0$ for a scalar $\lambda$. This means:

(A) $(A_1x + B_1y + C_1z + D_1) + \lambda(A_2x + B_2y + C_2z + D_2) = 0$.

(B) This equation represents a family of planes passing through the line of intersection of $P_1$ and $P_2$.

(C) The vector equation is $\vec{r} \cdot (\vec{n}_1 + \lambda \vec{n}_2) = d_1 + \lambda d_2$.

(D) If we know a point on the desired plane (not on the intersection line), we can find the value of $\lambda$.

Question 7. The direction cosines of the normal to the plane $Ax + By + Cz + D = 0$ are:

(A) Proportional to $(A, B, C)$.

(B) $(\frac{A}{\sqrt{A^2+B^2+C^2}}, \frac{B}{\sqrt{A^2+B^2+C^2}}, \frac{C}{\sqrt{A^2+B^2+C^2}})$.

(C) The same as the direction cosines of the vector $A\hat{i} + B\hat{j} + C\hat{k}$.

(D) Always positive.

Question 8. The equation of the plane passing through the point $(1,1,1)$ and having normal vector $2\hat{i} - \hat{j} + 3\hat{k}$ is:

(A) $2(x-1) - 1(y-1) + 3(z-1) = 0$.

(B) $2x - y + 3z - 4 = 0$.

(C) $\vec{r} \cdot (2\hat{i} - \hat{j} + 3\hat{k}) = (1\hat{i} + 1\hat{j} + 1\hat{k}) \cdot (2\hat{i} - \hat{j} + 3\hat{k}) = 2 - 1 + 3 = 4$.

(D) $\vec{r} \cdot (2\hat{i} - \hat{j} + 3\hat{k}) = 4$.

Question 9. Which of the following statements are true about a plane?

(A) A plane is a flat 2D surface extending infinitely.

(B) A plane is determined by three non-collinear points.

(C) A plane is determined by a point on the plane and a normal vector to the plane.

(D) The direction of the normal vector determines the orientation of the plane in space.

Question 10. The plane $2x - 3y + 4z + 5 = 0$ is parallel to which of the following planes?

(A) $2x - 3y + 4z = 10$

(B) $4x - 6y + 8z - 1 = 0$

(C) $\vec{r} \cdot (2\hat{i} - 3\hat{j} + 4\hat{k}) = 1$

(D) $2x - 3y + 4z = -5$

Question 11. The distance of a point $(x_1, y_1, z_1)$ from the plane $Ax+By+Cz+D=0$ is $\frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2+B^2+C^2}}$. Using this, the distance of the origin $(0,0,0)$ from the plane $x+2y-2z=6$ is:

(A) $\frac{|1(0)+2(0)-2(0)-6|}{\sqrt{1^2+2^2+(-2)^2}} = \frac{|-6|}{\sqrt{1+4+4}} = \frac{6}{3} = 2$

(B) 2

(C) A non-negative scalar value.

(D) The distance from the origin to the plane $\vec{r} \cdot (\hat{i} + 2\hat{j} - 2\hat{k}) = 6$.

Question 12. The plane $x+y+z=1$ makes intercepts on the axes. Which of the following are true?

(A) The x-intercept is 1.

(B) The y-intercept is 1.

(C) The z-intercept is 1.

(D) The equation in intercept form is $\frac{x}{1} + \frac{y}{1} + \frac{z}{1} = 1$.

Question 1. The angle between two lines is the angle between their direction vectors. If the direction vectors are $\vec{b}_1$ and $\vec{b}_2$, and $\theta$ is the angle between them, then $\cos \theta = \frac{|\vec{b}_1 \cdot \vec{b}_2|}{|\vec{b}_1||\vec{b}_2|}$. This formula gives the acute angle between the lines. Which of the following are true?

(A) The dot product $\vec{b}_1 \cdot \vec{b}_2$ is used.

(B) The magnitudes of the direction vectors are used.

(C) If $\vec{b}_1 \cdot \vec{b}_2 = 0$, the lines are perpendicular.

(D) If $\vec{b}_1 = k \vec{b}_2$ for some scalar $k$, the lines are parallel.

Question 2. The angle between two planes is the angle between their normal vectors. If the normal vectors are $\vec{n}_1$ and $\vec{n}_2$, and $\theta$ is the angle between the planes, then $\cos \theta = \frac{|\vec{n}_1 \cdot \vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$. Which of the following are true?

(A) The angle is the acute angle between the normals.

(B) If $\vec{n}_1 \cdot \vec{n}_2 = 0$, the planes are perpendicular.

(C) If $\vec{n}_1 = k \vec{n}_2$ for some scalar $k$, the planes are parallel or coincident.

(D) The formula uses the dot product of the normal vectors.

Question 3. The angle $\phi$ between a line with direction vector $\vec{b}$ and a plane with normal vector $\vec{n}$ is given by $\sin \phi = \frac{|\vec{b} \cdot \vec{n}|}{|\vec{b}||\vec{n}|}$. Which of the following are true?

(A) The angle between the line and the plane is the complement of the angle between the line and the normal to the plane.

(B) If $\vec{b} \cdot \vec{n} = 0$, the line is parallel to the plane (or lies in the plane).

(C) If $\vec{b}$ is parallel to $\vec{n}$ (i.e., $\vec{b} \times \vec{n} = \vec{0}$), the line is perpendicular to the plane.

(D) The formula gives the acute angle between the line and the plane.

Question 4. Consider the line $\frac{x-1}{2} = \frac{y-2}{-3} = \frac{z-3}{4}$ and the plane $2x + y - 3z = 5$. Which of the following are true?

(A) The direction vector of the line is $\vec{b} = 2\hat{i} - 3\hat{j} + 4\hat{k}$.

(B) The normal vector to the plane is $\vec{n} = 2\hat{i} + \hat{j} - 3\hat{k}$.

(C) $\vec{b} \cdot \vec{n} = (2)(2) + (-3)(1) + (4)(-3) = 4 - 3 - 12 = -11$.

(D) The lines are perpendicular because the dot product of their direction ratios is 0.

Question 5. Two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are parallel if:

(A) Their direction vectors are parallel.

(B) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ (assuming non-zero denominators).

(C) $a_1b_2 - a_2b_1 = 0, b_1c_2 - b_2c_1 = 0, c_1a_2 - c_2a_1 = 0$.

(D) The angle between them is $0^\circ$ or $180^\circ$.

Question 6. Two planes $A_1x + B_1y + C_1z + D_1 = 0$ and $A_2x + B_2y + C_2z + D_2 = 0$ are perpendicular if:

(A) Their normal vectors are perpendicular.

(B) $(A_1\hat{i} + B_1\hat{j} + C_1\hat{k}) \cdot (A_2\hat{i} + B_2\hat{j} + C_2\hat{k}) = 0$.

(C) $A_1A_2 + B_1B_2 + C_1C_2 = 0$.

(D) The angle between them is $90^\circ$.

Question 7. A line is perpendicular to a plane if:

(A) The direction vector of the line is parallel to the normal vector of the plane.

(B) The direction vector of the line is a scalar multiple of the normal vector of the plane.

(C) The angle between the line and the plane is $90^\circ$.

(D) The angle between the direction vector and the normal vector is $0^\circ$ or $180^\circ$.

Question 8. The angle between the line $\vec{r} = \vec{a} + \lambda \vec{b}$ and the plane $\vec{r} \cdot \vec{n} = d$ is given by $\phi$. If $\phi=0$, which of the following can be true?

(A) The line is parallel to the plane.

(B) The line lies in the plane.

(C) $\vec{b} \cdot \vec{n} = 0$.

(D) $\vec{a} \cdot \vec{n} = d$ and $\vec{b} \cdot \vec{n} = 0$ (line lies in the plane).

Question 9. The angle between the x-axis and the y-axis in 3D space is:

(A) $90^\circ$

(B) $\pi/2$ radians

(C) The angle between $\hat{i}$ and $\hat{j}$.

(D) Given by $\cos \theta = \frac{|\hat{i} \cdot \hat{j}|}{|\hat{i}||\hat{j}|} = \frac{0}{1 \cdot 1} = 0$.

Question 10. The angle between the plane $z=0$ (xy-plane) and the plane $x=0$ (yz-plane) is:

(A) $90^\circ$

(B) The angle between their normal vectors $\hat{k}$ and $\hat{i}$.

(C) The angle between the normal vectors $(0,0,1)$ and $(1,0,0)$.

(D) $\pi/2$ radians.

Question 11. If two lines are perpendicular, their direction vectors satisfy:

(A) $\vec{b}_1 \cdot \vec{b}_2 = 0$

(B) The angle between them is $90^\circ$.

(C) If their direction ratios are $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$, then $a_1a_2 + b_1b_2 + c_1c_2 = 0$.

(D) Their cross product is $\vec{0}$.

Question 1. The shortest distance between two skew lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$ is given by $d = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}$. This formula is valid if:

(A) The lines are non-parallel.

(B) The lines are non-intersecting.

(C) The lines are skew.

(D) $\vec{b}_1 \times \vec{b}_2 \neq \vec{0}$.

Question 2. The shortest distance between two parallel lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}$ is given by $d = \frac{|(\vec{a}_2 - \vec{a}_1) \times \vec{b}|}{|\vec{b}|}$. This formula uses:

(A) The vector connecting a point on the first line to a point on the second line ($\vec{a}_2 - \vec{a}_1$).

(B) The direction vector of the lines ($\vec{b}$).

(C) The magnitude of the cross product of $(\vec{a}_2 - \vec{a}_1)$ and $\vec{b}$.

(D) The magnitude of the direction vector $|\vec{b}|$.

Question 3. The distance of a point P$(x_1, y_1, z_1)$ from the plane $Ax + By + Cz + D = 0$ is $d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2+B^2+C^2}}$. This formula implies:

(A) The numerator is the absolute value of the result of substituting the point's coordinates into the plane equation.

(B) The denominator is the magnitude of the normal vector to the plane.

(C) If the point lies on the plane, the distance is 0.

(D) The formula is applicable even if the plane passes through the origin (when $D=0$).

Question 4. Skew lines are lines that are:

(A) In 3D space.

(B) Not parallel.

(C) Not intersecting.

(D) Coplanar.

Question 5. The distance between two parallel planes $Ax + By + Cz + D_1 = 0$ and $Ax + By + Cz + D_2 = 0$ is given by $d = \frac{|D_2 - D_1|}{\sqrt{A^2+B^2+C^2}}$. This formula is based on finding the distance of a point on one plane from the other plane. Which of the following are true?

(A) The normal vector $(A, B, C)$ is the same for both planes.

(B) A point on the first plane can be found by setting $x, y, z$ values (e.g., setting two variables to 0 and solving for the third, if possible).

(C) The distance is zero if $D_1 = D_2$ and the planes are identical.

(D) The formula can be derived from the point-to-plane distance formula.

Question 6. The distance of a point P from a line L can be found using vectors. If A is a point on the line and $\vec{b}$ is the direction vector of the line, the distance is the magnitude of the projection of $\vec{AP}$ onto a vector perpendicular to $\vec{b}$ in the plane containing P and L. Which of the following expressions can be used?

(A) $\frac{|\vec{AP} \times \vec{b}|}{|\vec{b}|}$

(B) $|\vec{AP}| \sin \theta$, where $\theta$ is the angle between $\vec{AP}$ and $\vec{b}$.

(C) $\sqrt{|\vec{AP}|^2 - (\text{Projection of } \vec{AP} \text{ on } \vec{b})^2}$

(D) $\sqrt{|\vec{AP}|^2 - \left(\frac{\vec{AP} \cdot \vec{b}}{|\vec{b}|}\right)^2}$

Question 7. The shortest distance between two lines is zero if and only if the lines are:

(A) Intersecting.

(B) Coincident.

(C) Coplanar.

(D) Parallel and distinct.

Question 8. Consider the distance of the point $(1,2,3)$ from the plane $z=0$. Which of the following are true?

(A) The plane is the xy-plane.

(B) The normal vector to the plane is $\hat{k}$.

(C) The equation of the plane is $0x + 0y + 1z - 0 = 0$, i.e., $z=0$. Using $z-0=0$ form, $A=0, B=0, C=1, D=0$.

(D) The distance is $\frac{|0(1) + 0(2) + 1(3) - 0|}{\sqrt{0^2+0^2+1^2}} = \frac{3}{1} = 3$.

Question 9. The lines $\vec{r} = \hat{i} + \lambda (\hat{j})$ and $\vec{r} = 2\hat{i} + \mu (\hat{j})$ are parallel. Which of the following are true about the shortest distance between them?

(A) $\vec{a}_1 = \hat{i}, \vec{a}_2 = 2\hat{i}, \vec{b} = \hat{j}$.

(B) $\vec{a}_2 - \vec{a}_1 = \hat{i}$.

(C) $(\vec{a}_2 - \vec{a}_1) \times \vec{b} = \hat{i} \times \hat{j} = \hat{k}$.

(D) Distance is $\frac{|\hat{k}|}{|\hat{j}|} = \frac{1}{1} = 1$.

Question 10. The shortest distance between the planes $x+y+z=1$ and $x+y+z=5$ is:

(A) The planes are parallel.

(B) Normal vector is $\hat{i} + \hat{j} + \hat{k}$.

(C) Distance is $\frac{|5-1|}{\sqrt{1^2+1^2+1^2}} = \frac{4}{\sqrt{3}}$.

(D) $\frac{4}{\sqrt{3}}$.

Question 11. If two lines intersect, their shortest distance is:

(A) 0

(B) The lines are coplanar.

(C) The scalar triple product $(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0$.

(D) The cross product $\vec{b}_1 \times \vec{b}_2 \neq \vec{0}$.