Completing Statements MCQs for Sub-Topics of Topic 12: Vectors & Three-Dimensional Geometry

Question 1. A quantity that is completely described by its magnitude alone is called a ______.

(A) Vector quantity

(B) Scalar quantity

(C) Tensor quantity

(D) Fundamental quantity

Question 2. A vector having magnitude equal to unity is known as a ______.

(A) Null vector

(B) Proper vector

(C) Unit vector

(D) Coinitial vector

Question 3. Two vectors $\vec{a}$ and $\vec{b}$ are equal if and only if they have the same magnitude and ______.

(A) Different directions

(B) The same direction

(C) The same initial point

(D) The same terminal point

Question 4. If two vectors $\vec{a}$ and $\vec{b}$ are collinear, then one vector can be expressed as a scalar multiple of the other, i.e., $\vec{a} = k\vec{b}$ for some scalar $k$, provided both vectors are ______.

(A) Unit vectors

(B) Zero vectors

(C) Non-zero vectors

(D) Equal vectors

Question 5. The vector sum of two vectors $\vec{a}$ and $\vec{b}$, represented by adjacent sides of a parallelogram, is given by the diagonal of the parallelogram originating from the ______.

(A) Terminal point of $\vec{a}$

(B) Terminal point of $\vec{b}$

(C) Initial point common to $\vec{a}$ and $\vec{b}$

(D) Midpoint of the parallelogram

Question 6. The subtraction of a vector $\vec{b}$ from a vector $\vec{a}$ is equivalent to adding vector $\vec{a}$ to the ______ vector of $\vec{b}$.

(A) Unit

(B) Zero

(C) Opposite

(D) Equal

Question 7. Multiplying a vector by a positive scalar changes its magnitude but preserves its ______.

(A) Initial point

(B) Direction

(C) Terminal point

(D) Components

Question 8. For any vector $\vec{a}$, $\vec{a} + \vec{0} = \vec{a}$ demonstrates the existence of a vector additive ______.

(A) Inverse

(B) Element

(C) Property

(D) Identity

Question 9. If two vectors $\vec{u}$ and $\vec{v}$ are coinitial, they share the same ______.

(A) Direction

(B) Magnitude

(C) Initial point

(D) Terminal point

Question 10. According to the triangle law of vector addition, if two vectors represent two sides of a triangle taken in order, their sum is represented by the third side taken in ______.

(A) The same order

(B) The reverse order

(C) Any order

(D) A proportional order

Question 1. The position vector of a point P$(x, y, z)$ with respect to the origin O is given by $\vec{OP} =$ ______.

(A) $(x, y, z)$

(B) $x+y+z$

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

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

Question 2. The vector joining point A with position vector $\vec{a}$ to point B with position vector $\vec{b}$ is represented by the vector ______.

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

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

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

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

Question 3. The magnitude of a vector $\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$ is calculated as ______.

(A) $|v_x| + |v_y| + |v_z|$

(B) $\sqrt{v_x^2 + v_y^2 + v_z^2}$

(C) $v_x^2 + v_y^2 + v_z^2$

(D) $(v_x, v_y, v_z)$

Question 4. Two vectors $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ are collinear if there exists a scalar $k$ such that $a_1 = kb_1, a_2 = kb_2, a_3 = kb_3$, provided that ______.

(A) $k=1$

(B) $k=-1$

(C) $k \neq 0$ and $\vec{b} \neq \vec{0}$

(D) $k$ is any real number

Question 5. A vector $\vec{v}$ is a linear combination of vectors $\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n$ if it can be written in the form $c_1\vec{v}_1 + c_2\vec{v}_2 + \dots + c_n\vec{v}_n$, where $c_1, c_2, \dots, c_n$ are ______.

(A) Vectors

(B) Scalars

(C) Unit vectors

(D) Zero vectors

Question 6. Three vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar if and only if one of the vectors can be expressed as a linear combination of the other two, provided the other two vectors are ______.

(A) Equal

(B) Perpendicular

(C) Collinear

(D) Non-collinear

Question 7. According to a theorem, any vector in a 2D plane can be uniquely expressed as a linear combination of two non-zero, non-collinear vectors in that plane, which form a ______.

(A) Unit set

(B) Orthogonal set

(C) Basis

(D) Null set

Question 8. The components of the vector $\vec{PQ}$, where P is $(x_1, y_1, z_1)$ and Q is $(x_2, y_2, z_2)$, are given by the ordered triplet ______.

(A) $(x_1-x_2, y_1-y_2, z_1-z_2)$

(B) $(x_1+x_2, y_1+y_2, z_1+z_2)$

(C) $(x_2-x_1, y_2-y_1, z_2-z_1)$

(D) $(|x_2-x_1|, |y_2-y_1|, |z_2-z_1|)$

Question 9. For a vector $\vec{v} = x\hat{i} + y\hat{j}$, its components are $x$ and $y$, and its magnitude is ______.

(A) $x+y$

(B) $\sqrt{x+y}$

(C) $x^2+y^2$

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

Question 10. If the vectors $\vec{a}$ and $\vec{b}$ are collinear, there exists a scalar $\lambda$ such that $\vec{a} = \lambda \vec{b}$. If $\lambda > 0$, they have the same direction; if $\lambda < 0$, they have ______ directions.

(A) Perpendicular

(B) Arbitrary

(C) Opposite

(D) Equal

Question 1. The scalar or dot product of two vectors $\vec{a}$ and $\vec{b}$ is defined as $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$, where $\theta$ is the angle between the vectors. The result of the dot product is always a ______.

(A) Vector quantity

(B) Scalar quantity

(C) Unit vector

(D) Zero vector

Question 2. If $\vec{a}$ and $\vec{b}$ are non-zero vectors, then $\vec{a} \cdot \vec{b} = 0$ if and only if the angle between $\vec{a}$ and $\vec{b}$ is ______ degrees.

(A) 0

(B) 45

(C) 90

(D) 180

Question 3. The scalar product $\vec{a} \cdot \vec{b}$ in terms of components, for $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, is given by ______.

(A) $(a_1+b_1)\hat{i} + (a_2+b_2)\hat{j} + (a_3+b_3)\hat{k}$

(B) $a_1b_1 + a_2b_2 + a_3b_3$

(C) $\sqrt{(a_1b_1)^2 + (a_2b_2)^2 + (a_3b_3)^2}$

(D) $(a_2b_3-a_3b_2)\hat{i} + \dots$

Question 4. The geometrical interpretation of the scalar projection of vector $\vec{a}$ on vector $\vec{b}$ is the length of the segment obtained by dropping a perpendicular from the tip of $\vec{a}$ onto the line containing $\vec{b}$, and its value is given by ______.

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

(B) $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$

(C) $|\vec{a}||\vec{b}|\cos\theta$

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

Question 5. One of the key applications of the dot product in physics is calculating the ______ done by a constant force.

(A) Velocity

(B) Acceleration

(C) Torque

(D) Work

Question 6. For any vector $\vec{a}$, the dot product of the vector with itself, $\vec{a} \cdot \vec{a}$, is equal to the ______ of its magnitude.

(A) Square root

(B) Square

(C) Reciprocal

(D) Cube

Question 7. The scalar product is commutative, meaning that for any two vectors $\vec{a}$ and $\vec{b}$, ______.

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

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

(C) $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|$

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

Question 8. If the angle between two non-zero vectors is acute, their dot product is ______.

(A) Zero

(B) Negative

(C) Positive

(D) Undefined

Question 9. The property $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$ is called the ______ property of scalar product over vector addition.

(A) Commutative

(B) Associative

(C) Distributive

(D) Identity

Question 10. For orthonormal basis vectors $\hat{i}, \hat{j}, \hat{k}$, the dot product $\hat{i} \cdot \hat{j}$ is equal to ______.

(A) 1

(B) 0

(C) $\hat{k}$

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

Question 1. The magnitude of the vector or cross product of two vectors $\vec{a}$ and $\vec{b}$ is defined as $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$, where $\theta$ is the angle between the vectors. The result of the cross product is always a ______.

(A) Scalar quantity

(B) Vector quantity

(C) Unit scalar

(D) Zero scalar

Question 2. If $\vec{a}$ and $\vec{b}$ are non-zero vectors, then $\vec{a} \times \vec{b} = \vec{0}$ if and only if $\vec{a}$ and $\vec{b}$ are ______.

(A) Perpendicular

(B) Equal

(C) Collinear or parallel

(D) Coinitial

Question 3. The direction of the vector $\vec{a} \times \vec{b}$ is perpendicular to the plane containing $\vec{a}$ and $\vec{b}$, and its specific orientation is determined by the ______ rule.

(A) Left-hand

(B) Right-hand

(C) Triangle

(D) Parallelogram

Question 4. The magnitude of the cross product $|\vec{a} \times \vec{b}|$ geometrically represents the area of the ______ formed by adjacent sides $\vec{a}$ and $\vec{b}$.

(A) Triangle

(B) Square

(C) Parallelogram

(D) Rectangle

Question 5. One of the key applications of the cross product in physics is calculating the ______ produced by a force about a point.

(A) Work

(B) Power

(C) Torque

(D) Energy

Question 6. The cross product is anti-commutative, meaning that for any two vectors $\vec{a}$ and $\vec{b}$, ______.

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

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

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

(D) $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|$

Question 7. For orthonormal basis vectors $\hat{i}, \hat{j}, \hat{k}$, the cross product $\hat{i} \times \hat{j}$ is equal to ______.

(A) 1

(B) 0

(C) $\hat{k}$

(D) $-\hat{k}$

Question 8. If the angle between two non-zero vectors is $90^\circ$, the magnitude of their cross product is equal to the ______ of their magnitudes.

(A) Sum

(B) Difference

(C) Product

(D) Ratio

Question 9. The vector product of a vector with itself is always the ______ vector.

(A) Unit

(B) Null (Zero)

(C) Opposite

(D) Same

Question 10. The area of a triangle with adjacent sides $\vec{a}$ and $\vec{b}$ is given by ______.

(A) $|\vec{a} \times \vec{b}|$

(B) $\frac{1}{2}|\vec{a} \times \vec{b}|$

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

(D) $\frac{1}{2}|\vec{a} \cdot \vec{b}|$

Question 1. The scalar triple product of three vectors $\vec{a}, \vec{b}, \vec{c}$ is defined as $\vec{a} \cdot (\vec{b} \times \vec{c})$. The result of this operation is always a ______.

(A) Vector quantity

(B) Scalar quantity

(C) Zero vector

(D) Unit vector

Question 2. The absolute value of the scalar triple product $|[\vec{a}, \vec{b}, \vec{c}]|$ represents the volume of the ______ formed by the vectors $\vec{a}, \vec{b}, \vec{c}$ as adjacent edges.

(A) Triangle

(B) Parallelogram

(C) Tetrahedron

(D) Parallelepiped

Question 3. Three vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar if and only if their scalar triple product $[\vec{a}, \vec{b}, \vec{c}]$ is equal to ______.

(A) 1

(B) -1

(C) 0

(D) Infinity

Question 4. Interchanging any two vectors in a scalar triple product changes its ______.

(A) Magnitude

(B) Sign

(C) Value to zero

(D) Dimension

Question 5. If any two of the three vectors in a scalar triple product are parallel or identical, the value of the product is ______.

(A) 1

(B) -1

(C) 0

(D) Non-zero

Question 6. The volume of a tetrahedron with coterminous edges given by vectors $\vec{a}, \vec{b}, \vec{c}$ is equal to ______ the absolute value of their scalar triple product.

(A) Half

(B) One-third

(C) One-sixth

(D) Equal to

Question 7. The scalar triple product $[\vec{a}, \vec{b}, \vec{c}]$ can be calculated as the determinant formed by the ______ of the three vectors.

(A) Magnitudes

(B) Directions

(C) Components

(D) Initial points

Question 8. If $[\vec{u}, \vec{v}, \vec{w}] \neq 0$, then the vectors $\vec{u}, \vec{v}, \vec{w}$ are said to be ______.

(A) Collinear

(B) Coplanar

(C) Mutually perpendicular

(D) Non-coplanar

Question 9. The property $\vec{a} \cdot (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}) \cdot \vec{c}$ allows for the interchange of the dot and cross in the scalar triple product, provided the ______ of the vectors are maintained.

(A) Magnitudes

(B) Components

(C) Cyclic order

(D) Angles

Question 10. If the scalar triple product $[\vec{a}, \vec{b}, \vec{c}]$ is positive, the vectors $\vec{a}, \vec{b}, \vec{c}$ form a ______ handed system.

(A) Left

(B) Right

(C) Neutral

(D) Zero

Question 1. The position vector of a point R that divides the line segment joining points A($\vec{a}$) and B($\vec{b}$) internally in the ratio $m:n$ is given by $\vec{r} =$ ______.

(A) $\frac{m\vec{a} + n\vec{b}}{m+n}$

(B) $\frac{n\vec{a} + m\vec{b}}{m+n}$

(C) $\frac{m\vec{b} - n\vec{a}}{m-n}$

(D) $\frac{n\vec{a} - m\vec{b}}{n-m}$

Question 2. The position vector of the midpoint of the line segment joining points A and B with position vectors $\vec{a}$ and $\vec{b}$ is given by $\vec{r} =$ ______.

(A) $\frac{\vec{a} + \vec{b}}{2}$

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

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

(D) $\frac{2\vec{a} + \vec{b}}{3}$

Question 3. The position vector of a point R that divides the line segment joining points A($\vec{a}$) and B($\vec{b}$) externally in the ratio $m:n$ is given by $\vec{r} =$ ______ (assuming $m \neq n$).

(A) $\frac{n\vec{a} + m\vec{b}}{m+n}$

(B) $\frac{m\vec{a} - n\vec{b}}{m-n}$

(C) $\frac{m\vec{b} - n\vec{a}}{m-n}$

(D) $\frac{n\vec{b} - m\vec{a}}{n-m}$

Question 4. If A, B, C are vertices of a triangle with position vectors $\vec{a}, \vec{b}, \vec{c}$ respectively, the position vector of its centroid G is given by $\vec{g} =$ ______.

(A) $\frac{\vec{a} + \vec{b}}{2}$

(B) $\frac{\vec{a} + \vec{b} + \vec{c}}{3}$

(C) $\vec{a} + \vec{b} + \vec{c}$

(D) $\frac{1}{2}(\vec{a} + \vec{b} + \vec{c})$

Question 5. The section formula is used to find the position vector of a point that divides a line segment in a given ratio, which implies that the points involved (the endpoints and the division point) must be ______.

(A) Perpendicular

(B) Coplanar

(C) Collinear

(D) Orthogonal

Question 6. If point R divides the line segment AB such that $\vec{AR} = k \vec{RB}$ for a positive scalar $k$, then R divides AB ______.

(A) Externally

(B) Internally

(C) At the midpoint

(D) In a ratio $k:1$ externally

Question 7. The position vector of the centroid of a tetrahedron with vertices having position vectors $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ is given by ______.

(A) $\frac{\vec{a} + \vec{b} + \vec{c}}{3}$

(B) $\frac{\vec{a} + \vec{b} + \vec{c} + \vec{d}}{4}$

(C) $\vec{a} + \vec{b} + \vec{c} + \vec{d}$

(D) $\frac{1}{2}(\vec{a} + \vec{b} + \vec{c} + \vec{d})$

Question 8. If point R divides the line segment AB externally in the ratio $m:n$, where $m>n$, R lies on the extension of AB ______.

(A) Between A and B

(B) Beyond B

(C) Beyond A

(D) At the midpoint

Question 9. The midpoint formula is a special case of the internal section formula where the ratio of division is ______.

(A) $m:n = 1:0$

(B) $m:n = 0:1$

(C) $m:n = 1:1$

(D) $m:n = 2:1$

Question 10. If point C divides line segment AB externally in the ratio $1:2$ (A to B), then C lies on the line AB extended beyond point ______.

(A) B

(B) A

(C) The midpoint

(D) Neither A nor B

Question 1. In a three-dimensional coordinate system, the distance of a point $(x, y, z)$ from the origin $(0, 0, 0)$ is given by ______.

(A) $|x| + |y| + |z|$

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

(C) $x^2+y^2+z^2$

(D) $\sqrt{x+y+z}$

Question 2. If a line makes angles $\alpha, \beta, \gamma$ with the positive directions of the x, y, and z axes respectively, then $\cos \alpha, \cos \beta, \cos \gamma$ are called the ______ of the line.

(A) Direction ratios

(B) Direction angles

(C) Direction cosines

(D) Components

Question 3. The sum of the squares of the direction cosines of any line in 3D space is always equal to ______.

(A) 0

(B) 1

(C) -1

(D) 3

Question 4. The vector equation of a straight line passing through a point with position vector $\vec{a}$ and parallel to a non-zero vector $\vec{b}$ is $\vec{r} =$ ______.

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

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

(C) $\vec{a} + \lambda \vec{b}$

(D) $\lambda (\vec{a} + \vec{b})$

Question 5. The Cartesian equation of a straight line passing through $(x_1, y_1, z_1)$ and having direction ratios $a, b, c$ is given by ______.

(A) $a(x-x_1) + b(y-y_1) + c(z-z_1) = 0$

(B) $\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$

(C) $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$

(D) $\sqrt{(x-x_1)^2 + (y-y_1)^2 + (z-z_1)^2}$

Question 6. The vector equation of a line passing through two points with position vectors $\vec{a}$ and $\vec{b}$ is $\vec{r} =$ ______.

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

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

(C) $\lambda \vec{a} + \mu \vec{b}$ where $\lambda + \mu = 1$

(D) Both (B) and (C)

Question 7. If the direction cosines of a line are proportional to $(a, b, c)$, then $(a, b, c)$ are called its ______.

(A) Direction angles

(B) Direction vectors

(C) Direction ratios

(D) Components

Question 8. The Cartesian coordinates of a point on the line $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} = \lambda$ are given by $x = x_1 + \lambda a, y = y_1 + \lambda b, z = z_1 + \lambda c$, where $\lambda$ is the ______.

(A) Direction ratio

(B) Direction cosine

(C) Parameter

(D) Magnitude

Question 9. The distance of the point $(x_1, y_1, z_1)$ from the xy-plane is ______.

(A) $|x_1|$

(B) $|y_1|$

(C) $|z_1|$

(D) $\sqrt{x_1^2+y_1^2}$

Question 10. A line parallel to the x-axis has direction ratios proportional to ______.

(A) $(0, 1, 1)$

(B) $(1, 0, 0)$

(C) $(0, 1, 0)$

(D) $(1, 1, 0)$

Question 1. The vector equation of a plane perpendicular to a unit vector $\hat{n}$ and at a distance $d$ from the origin is $\vec{r} \cdot \hat{n} =$ ______.

(A) 0

(B) 1

(C) $d$

(D) $|\vec{r}||\hat{n}|$

Question 2. In the Cartesian equation of a plane $Ax + By + Cz + D = 0$, the vector $A\hat{i} + B\hat{j} + C\hat{k}$ is a ______ vector to the plane.

(A) Direction

(B) Position

(C) Normal

(D) Unit

Question 3. The equation of a plane passing through a point with position vector $\vec{a}$ and perpendicular to vector $\vec{n}$ is given by $(\vec{r} - \vec{a}) \cdot \vec{n} =$ ______.

(A) $|\vec{r} - \vec{a}||\vec{n}|$

(B) 1

(C) $\vec{0}$

(D) 0

Question 4. The intercept form of the equation of a plane is $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$, where $a, b, c$ are the non-zero intercepts on the ______ axes respectively.

(A) xyz

(B) Normal

(C) Coordinate

(D) Planar

Question 5. The distance of the plane $Ax + By + Cz + D = 0$ from the origin is given by the formula ______.

(A) $\frac{|D|}{A+B+C}$

(B) $\frac{|D|}{\sqrt{A^2+B^2+C^2}}$

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

(D) $|D|$

Question 6. A plane passing through the origin has its constant term $D$ in the equation $Ax + By + Cz + D = 0$ equal to ______.

(A) 1

(B) -1

(C) 0

(D) Any real number

Question 7. The equation of any plane passing through the intersection of two planes $P_1$ and $P_2$ is given by $P_1 + \lambda P_2 = 0$, where $\lambda$ is a ______.

(A) Vector

(B) Plane

(C) Scalar

(D) Line

Question 8. A plane is uniquely determined by three ______ points.

(A) Collinear

(B) Coplanar

(C) Non-collinear

(D) Mutually perpendicular

Question 9. The Cartesian equation of the plane $\vec{r} \cdot (A\hat{i} + B\hat{j} + C\hat{k}) = D'$ is ______.

(A) $A x + B y + C z = D'$

(B) $A x + B y + C z + D' = 0$

(C) $x/A + y/B + z/C = D'$

(D) $A x + B y + C z = 0$

Question 10. The equation of the yz-plane is $x=0$. Its normal vector can be taken as ______.

(A) $\hat{j}$

(B) $\hat{k}$

(C) $\hat{i}$

(D) $\hat{j} + \hat{k}$

Question 1. The angle between two lines with direction vectors $\vec{b}_1$ and $\vec{b}_2$ is determined by the formula $\cos\theta = \frac{|\vec{b}_1 \cdot \vec{b}_2|}{|\vec{b}_1||\vec{b}_2|}$, which gives the ______ angle between the lines.

(A) Obtuse

(B) Straight

(C) Acute

(D) Reflex

Question 2. Two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are perpendicular if the dot product of their direction vectors is zero, which means $a_1a_2 + b_1b_2 + c_1c_2 =$ ______.

(A) 1

(B) -1

(C) 0

(D) A non-zero value

Question 3. The angle between two planes is equal to the angle between their ______ vectors.

(A) Direction

(B) Position

(C) Normal

(D) Skew

Question 4. Two planes $A_1x + B_1y + C_1z + D_1 = 0$ and $A_2x + B_2y + C_2z + D_2 = 0$ are parallel if their normal vectors are parallel, which means their coefficients are proportional: $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}$ (assuming denominators are non-zero). If this ratio is equal to $\frac{D_1}{D_2}$, the planes are ______.

(A) Perpendicular

(B) Intersecting

(C) Skew

(D) Coincident

Question 5. The angle $\phi$ between a line with direction vector $\vec{b}$ and a plane with normal vector $\vec{n}$ is related to the angle $\theta$ between $\vec{b}$ and $\vec{n}$ by the formula $\phi + \theta =$ ______ degrees (assuming acute angle for line-plane).

(A) 0

(B) 45

(C) 90

(D) 180

Question 6. A line is parallel to a plane if its direction vector is ______ to the normal vector of the plane.

(A) Parallel

(B) Perpendicular

(C) Equal

(D) A scalar multiple

Question 7. A line is perpendicular to a plane if its direction vector is ______ to the normal vector of the plane.

(A) Perpendicular

(B) Parallel

(C) Orthogonal

(D) The zero vector

Question 8. If two lines are parallel, the angle between their direction vectors is either $0^\circ$ or ______ degrees.

(A) 45

(B) 90

(C) 180

(D) 270

Question 9. The angle between the xy-plane and the z-axis is ______ degrees.

(A) 0

(B) 45

(C) 90

(D) 180

Question 10. If two planes are perpendicular, the angle between their normal vectors is ______ degrees.

(A) 0

(B) 45

(C) 90

(D) 180

Question 1. The shortest distance between two skew lines is the length of the unique line segment which is perpendicular to ______.

(A) The first line only

(B) The second line only

(C) Both lines

(D) The line joining their starting points

Question 2. The formula for the shortest distance between two parallel lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}$ involves the magnitude of the cross product of $(\vec{a}_2 - \vec{a}_1)$ and $\vec{b}$, divided by the magnitude of ______.

(A) $\vec{a}_2 - \vec{a}_1$

(B) $\vec{b}$

(C) $(\vec{a}_2 - \vec{a}_1) \times \vec{b}$

(D) $\lambda \vec{b} + \mu \vec{b}$

Question 3. The distance of a point P$(x_1, y_1, z_1)$ from the plane $Ax + By + Cz + D = 0$ is given by the formula involving the absolute value of $Ax_1 + By_1 + Cz_1 + D$ divided by the magnitude of the plane's ______ vector.

(A) Direction

(B) Position

(C) Normal

(D) Unit

Question 4. If the shortest distance between two lines is zero, the lines are either intersecting or ______.

(A) Skew

(B) Parallel

(C) Perpendicular

(D) Coincident

Question 5. Skew lines are lines in 3D space that are neither parallel nor ______.

(A) Collinear

(B) Coplanar

(C) Perpendicular

(D) Intersecting

Question 6. The distance between two parallel planes $Ax + By + Cz + D_1 = 0$ and $Ax + By + Cz + D_2 = 0$ is given by the formula involving the absolute difference of $D_1$ and $D_2$ divided by ______.

(A) $\sqrt{D_1^2+D_2^2}$

(B) $|D_1-D_2|$

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

(D) $(A+B+C)$

Question 7. The distance of a point P from a line L is the length of the perpendicular segment from P to L, which meets the line at a point Q such that the vector $\vec{PQ}$ is ______ to the line L.

(A) Parallel

(B) Collinear

(C) Perpendicular

(D) Equal

Question 8. If two planes are coincident, the distance between them is ______.

(A) 1

(B) Finite and non-zero

(C) Infinite

(D) Zero

Question 9. The distance of the origin $(0,0,0)$ from the plane $Ax+By+Cz+D=0$ is obtained by setting $(x_1,y_1,z_1)$ to $(0,0,0)$ in the point-to-plane distance formula, resulting in $\frac{|D|}{\sqrt{A^2+B^2+C^2}}$, where $|D|$ is the absolute value of the ______ term.

(A) x-coefficient

(B) y-coefficient

(C) z-coefficient

(D) Constant

Question 10. For two lines in 3D space, the shortest distance is 0 if and only if they are ______.

(A) Skew

(B) Parallel

(C) Intersecting or coincident

(D) Perpendicular