Negative Questions MCQs for Sub-Topics of Topic 12: Vectors & Three-Dimensional Geometry

Question 1. Which of the following is NOT a scalar quantity?

(A) Temperature

(B) Electric current

(C) Work

(D) Electric field

Question 2. Which statement about a unit vector is FALSE?

(A) Its magnitude is 1.

(B) It specifies a direction.

(C) The unit vector in the direction of $\vec{v}$ is $\frac{\vec{v}}{|\vec{v}|}$.

(D) It can be the zero vector.

Question 3. Which pair of vectors are NOT collinear?

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

(B) $3\hat{i} - \hat{j}$ and $-6\hat{i} + 2\hat{j}$

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

(D) $\vec{a}$ and $k\vec{a}$ (for $k \neq 0$)

Question 4. Which of the following is NOT a property of vector addition?

(A) Commutativity

(B) Associativity

(C) Existence of inverse

(D) Distributivity over scalar multiplication

Question 5. If $\vec{a}$ and $\vec{b}$ are equal vectors, which statement is NOT true?

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

(B) They have the same direction.

(C) They represent the same displacement.

(D) They must have the same initial point.

Question 6. Which statement about scalar multiplication is INCORRECT?

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

(B) $(k_1 k_2)\vec{a} = k_1(k_2\vec{a})$

(C) $1 \cdot \vec{a} = \vec{a}$

(D) $k\vec{a}$ is always in the same direction as $\vec{a}$.

Question 7. Which of the following does NOT represent the vector sum of two vectors $\vec{a}$ and $\vec{b}$?

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

(B) The diagonal of a parallelogram with adjacent sides $\vec{a}$ and $\vec{b}$ starting from the same initial point.

(C) The third side of a triangle with sides $\vec{a}$ and $\vec{b}$ taken in order.

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

Question 8. If $k\vec{a} = \vec{0}$, where $k$ is a scalar and $\vec{a}$ is a vector, which is NOT necessarily true?

(A) $k = 0$

(B) $\vec{a} = \vec{0}$

(C) Either $k=0$ or $\vec{a} = \vec{0}$ (or both).

(D) $|\vec{a}| = 1$

Question 9. Which statement about the zero vector $\vec{0}$ is FALSE?

(A) It has zero magnitude.

(B) Its direction is indeterminate.

(C) For any vector $\vec{v}$, $\vec{v} - \vec{v} = \vec{0}$.

(D) For any non-zero scalar $k$, $k\vec{0}$ is a non-zero vector.

Question 10. Which of the following operations on vectors does NOT result in a vector?

(A) Sum of two vectors.

(B) Difference of two vectors.

(C) Product of a scalar and a vector.

(D) Magnitude of a vector.

Question 11. If $\vec{a}$ and $\vec{b}$ are two non-zero vectors, which inequality is NOT always true?

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

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

(C) $|\vec{a} + \vec{b}| = |\vec{a}| + |\vec{b}|$ (unless $\vec{a}$ and $\vec{b}$ are in the same direction)

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

Question 12. Which type of vector is NOT defined based on its position relative to other vectors or points?

(A) Coinitial vector

(B) Collinear vector

(C) Unit vector

(D) Equal vector

Question 1. Which is NOT the position vector of point P(x,y,z) with respect to the origin O?

(A) $\vec{OP}$

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

(C) $(x, y, z)$ (as a vector)

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

Question 2. Points A has position vector $\vec{a}$ and B has position vector $\vec{b}$. Which is NOT a correct representation of the vector $\vec{AB}$?

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

(B) A vector starting at A and ending at B.

(C) The displacement vector from A to B.

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

Question 3. Which formula for the magnitude of a vector is INCORRECT?

(A) $|\vec{v}| = \sqrt{v_x^2 + v_y^2}$ for $\vec{v} = v_x\hat{i} + v_y\hat{j}$

(B) $|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$ for $\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$

(C) $|\vec{AB}| = \sqrt{(x_B-x_A)^2 + (y_B-y_A)^2 + (z_B-z_A)^2}$ for A$(x_A,y_A,z_A)$, B$(x_B,y_B,z_B)$

(D) $|\vec{v}| = v_x + v_y + v_z$

Question 4. If two non-zero vectors $\vec{a}$ and $\vec{b}$ in component form are collinear, which is FALSE?

(A) $\vec{a} = k\vec{b}$ for some non-zero scalar $k$.

(B) The ratio of their corresponding components is constant.

(C) Their cross product is the zero vector.

(D) Their dot product is zero.

Question 5. Which is NOT a linear combination of vectors $\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n$?

(A) $c_1\vec{v}_1 + c_2\vec{v}_2 + \dots + c_n\vec{v}_n$ for scalars $c_i$

(B) $\vec{v}_1 + \vec{v}_2$

(C) $2\vec{v}_1 - 3\vec{v}_n$

(D) $|\vec{v}_1|\vec{v}_1 + |\vec{v}_2|\vec{v}_2$ (unless $|\vec{v}_i|$ are scalars)

Question 6. If vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar by components (i.e., their scalar triple product is zero), which is NOT necessarily true?

(A) They lie in the same plane.

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

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

(D) They are linearly dependent.

Question 7. Which statement about a basis in 2D (Theorem on Two Non-Zero Non-Collinear Vectors) is INCORRECT?

(A) Any two non-zero, non-collinear vectors in 2D form a basis.

(B) Any vector in the 2D plane can be uniquely expressed as a linear combination of the basis vectors.

(C) The standard basis vectors $\hat{i}, \hat{j}$ are the only possible basis vectors.

(D) The basis vectors are linearly independent.

Question 8. Which set of vectors cannot be a basis for 3D space?

(A) $\{\hat{i}, \hat{j}, \hat{k}\}$

(B) $\{2\hat{i}, 3\hat{j}, 4\hat{k}\}$

(C) $\{\hat{i} + \hat{j}, \hat{j} + \hat{k}, \hat{k} + \hat{i}\}$

(D) $\{\hat{i}, 2\hat{i}, 3\hat{k}\}$

Question 9. Given $\vec{a} = a_1\hat{i} + a_2\hat{j}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j}$. Which component operation is INCORRECT?

(A) $\vec{a} + \vec{b} = (a_1+b_1)\hat{i} + (a_2+b_2)\hat{j}$

(B) $\vec{a} - \vec{b} = (a_1-b_1)\hat{i} + (a_2-b_2)\hat{j}$

(C) $k\vec{a} = (ka_1)\hat{i} + (ka_2)\hat{j}$

(D) $\vec{a} \cdot \vec{b} = (a_1b_1)\hat{i} + (a_2b_2)\hat{j}$

Question 10. Points A, B, C have position vectors $\vec{a}, \vec{b}, \vec{c}$. If A, B, C are collinear, which statement about $\vec{AB}$ and $\vec{AC}$ is FALSE?

(A) $\vec{AB}$ is parallel to $\vec{AC}$.

(B) $\vec{AC} = k \vec{AB}$ for some scalar $k$.

(C) The ratio of corresponding components of $\vec{AB}$ and $\vec{AC}$ is constant.

(D) $|\vec{AB}| + |\vec{BC}| = |\vec{AC}|$ always holds (assuming B is between A and C).

Question 1. Which of the following is NOT a correct property of the scalar (dot) product?

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

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

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

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

Question 2. Which result is INCORRECT for dot products of standard basis vectors?

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

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

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

(D) $\hat{i} \cdot \hat{j} = 1$

Question 3. For non-zero vectors $\vec{a}$ and $\vec{b}$, if $\vec{a} \cdot \vec{b} \neq 0$, which is NOT true?

(A) The vectors are not perpendicular.

(B) The angle between them is not $90^\circ$.

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

(D) The angle between them is either acute or obtuse.

Question 4. Which is NOT a correct formula related to projection using the dot product?

(A) Scalar projection of $\vec{a}$ on $\vec{b}$: $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$

(B) Vector projection of $\vec{a}$ on $\vec{b}$: $(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2})\vec{b}$

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

(D) Magnitude of scalar projection: $|\vec{a}|\cos\theta$ (where $\theta$ is angle between $\vec{a}$ and $\vec{b}$)

Question 5. Which of the following is NOT an application of the dot product?

(A) Calculating work done by a constant force.

(B) Finding the angle between two vectors.

(C) Determining if two vectors are perpendicular.

(D) Finding a vector perpendicular to two given vectors.

Question 6. If non-zero vectors $\vec{a}$ and $\vec{b}$ satisfy $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|$, which statement is FALSE?

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

(B) $\vec{a}$ and $\vec{b}$ are parallel and in the same direction.

(C) $\vec{a} = k\vec{b}$ for some scalar $k > 0$.

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

Question 7. Which statement about the sign of the dot product and the angle between non-zero vectors is INCORRECT?

(A) If $\vec{a} \cdot \vec{b} > 0$, the angle is acute.

(B) If $\vec{a} \cdot \vec{b} < 0$, the angle is obtuse.

(C) If $\vec{a} \cdot \vec{b} = 0$, the angle is $90^\circ$.

(D) If the angle is $180^\circ$, $\vec{a} \cdot \vec{b}$ is positive.

Question 8. Given $\vec{a} = a_1\hat{i} + a_2\hat{j}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j}$. Which component calculation of the dot product is WRONG?

(A) $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2$

(B) $\hat{i} \cdot \vec{a} = a_1$

(C) $\vec{a} \cdot \hat{j} = a_2$

(D) $\vec{a} \cdot \vec{b} = (a_1+b_1)(a_2+b_2)$

Question 9. If non-zero vectors $\vec{a}$ and $\vec{b}$ are perpendicular, which is NOT implied?

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

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

(C) Their cross product is the zero vector.

(D) The projection of $\vec{a}$ on $\vec{b}$ is 0.

Question 10. Which inequality does NOT correctly relate the magnitudes and the dot product of two vectors $\vec{a}$ and $\vec{b}$?

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

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

Question 1. Which of the following is NOT a correct property of the vector (cross) product?

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

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

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

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

Question 2. Which result is INCORRECT for cross products of standard basis vectors?

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

Question 3. For non-zero vectors $\vec{a}$ and $\vec{b}$, if $\vec{a} \times \vec{b} = \vec{0}$, which is NOT necessarily true?

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

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

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

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

Question 4. Which is NOT a correct interpretation of $|\vec{a} \times \vec{b}|$?

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

(B) Twice the area of the triangle with 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) The scalar triple product of $\vec{a}, \vec{b}$ and a third vector.

Question 5. Which of the following is NOT an application of the cross product?

(A) Finding the area of a triangle in 3D space.

(B) Calculating torque.

(C) Finding a vector perpendicular to a plane.

(D) Calculating work done by a force.

Question 6. Which statement about the direction of $\vec{a} \times \vec{b}$ is FALSE?

(A) It is perpendicular to both $\vec{a}$ and $\vec{b}$.

(B) It is along the normal to the plane containing $\vec{a}$ and $\vec{b}$.

(C) Its direction is given by the right-hand rule.

(D) It is always in the direction of $\vec{a}$.

Question 7. Given $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$. Which determinant formula for the cross product $\vec{a} \times \vec{b}$ is INCORRECT?

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

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

Question 8. If non-zero vectors $\vec{a}$ and $\vec{b}$ satisfy $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|$, which is FALSE?

(A) The angle between $\vec{a}$ and $\vec{b}$ is $90^\circ$.

(B) $\sin\theta = 1$.

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

(D) $\vec{a} \cdot \vec{b} \neq 0$

Question 9. Which pair of non-zero vectors would NOT have a zero cross product?

(A) $\vec{a}$ and $2\vec{a}$

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

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

(D) $\hat{i} + \hat{j}$ and $2\hat{i} + 2\hat{j}$

Question 10. Which distributive property for the cross product is INCORRECT?

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

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

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

(D) $(\vec{a} + \vec{b}) \times \vec{c} = \vec{a} \times \vec{c} + \vec{b} \times \vec{c}$ is equivalent to $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$ due to anti-commutativity.

Question 1. Which is NOT a correct definition or representation of the scalar triple product of vectors $\vec{a}, \vec{b}, \vec{c}$?

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

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

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

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

Question 2. Which property is FALSE for the scalar triple product $[\vec{a}, \vec{b}, \vec{c}]$?

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

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

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

(D) $[\vec{a}+\vec{d}, \vec{b}, \vec{c}] = [\vec{a}, \vec{b}, \vec{c}] \cdot [\vec{d}, \vec{b}, \vec{c}]$

Question 3. If the scalar triple product $[\vec{a}, \vec{b}, \vec{c}] \neq 0$, which is NOT true?

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

(B) The volume of the parallelepiped formed by $\vec{a}, \vec{b}, \vec{c}$ is non-zero.

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

(D) The vectors can form a basis for 3D space.

Question 4. Which is NOT a correct geometric interpretation of $|[\vec{a}, \vec{b}, \vec{c}]|$?

(A) Volume of the parallelepiped with coterminous edges $\vec{a}, \vec{b}, \vec{c}$.

(B) Absolute value of $\vec{a} \cdot (\vec{b} \times \vec{c})$.

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

(D) The area of the triangle formed by $\vec{a}, \vec{b}$.

Question 5. Which condition does NOT imply that three vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar?

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

(B) One vector is the zero vector.

(C) Two vectors are parallel.

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

Question 6. If two vectors in the scalar triple product are identical, e.g., $[\vec{a}, \vec{a}, \vec{b}]$, which is FALSE?

(A) The value of the STP is 0.

(B) The vectors are coplanar.

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

(D) The vectors $\vec{a}, \vec{a}, \vec{b}$ form a basis for 3D space (unless $\vec{a}=\vec{0}$).

Question 7. Which cyclic permutation does NOT result in the same scalar triple product value as $[\vec{u}, \vec{v}, \vec{w}]$?

(A) $[\vec{v}, \vec{w}, \vec{u}]$

(B) $[\vec{w}, \vec{u}, \vec{v}]$

(C) $[\vec{u}, \vec{w}, \vec{v}]$

(D) $\vec{v} \cdot (\vec{w} \times \vec{u})$

Question 8. Which statement about the sign of the scalar triple product $[\vec{a}, \vec{b}, \vec{c}]$ is INCORRECT?

(A) If $[\vec{a}, \vec{b}, \vec{c}] > 0$, the vectors form a right-handed system.

(B) If $[\vec{a}, \vec{b}, \vec{c}] < 0$, the vectors form a left-handed system.

(C) The sign indicates the relative orientation of the vectors.

(D) The sign determines the volume of the parallelepiped.

Question 9. Given $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$, $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, $\vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$. Which calculation using determinants for $[\vec{a}, \vec{b}, \vec{c}]$ is WRONG?

(A) $\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$

(B) $\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$

(C) $\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} \cdot (c_1\hat{i} + c_2\hat{j} + c_3\hat{k})$

(D) $(a_1b_2 - a_2b_1)c_3 + (a_2b_3 - a_3b_2)c_1 + (a_3b_1 - a_1b_3)c_2$

Question 10. If $[\vec{a}, \vec{b}, \vec{c}] = 10$, what is NOT necessarily the volume of the parallelepiped formed by $\vec{a}, \vec{b}, \vec{c}$?

(A) 10

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

(C) The absolute value of the scalar triple product.

(D) $-10$

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 given by $\vec{r}$. Which is NOT the correct formula for $\vec{r}$?

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

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

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

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

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 given by $\vec{r}$. Which is NOT the correct formula for $\vec{r}$?

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

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

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

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

Question 3. Which point on the line segment AB (with position vectors $\vec{a}, \vec{b}$) does NOT correspond to dividing it in the ratio 1:1?

(A) The midpoint of AB.

(B) The point R with position vector $\frac{\vec{a} + \vec{b}}{2}$.

(C) The point dividing BA internally in the ratio 1:1.

(D) The point dividing AB externally in the ratio 1:1.

Question 4. If A, B, C are vertices of a triangle with position vectors $\vec{a}, \vec{b}, \vec{c}$, which is NOT the formula for the position vector of its centroid G?

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

(B) The point dividing the median from A (to midpoint of BC) in ratio 2:1.

(C) $\frac{1}{3}(\vec{a} + \vec{b} + \vec{c})$

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

Question 5. Which statement about external division of a line segment AB by a point R in the ratio $m:n$ is INCORRECT?

(A) R lies on the line AB but outside the segment AB.

(B) The formula $\vec{r} = \frac{m\vec{b} - n\vec{a}}{m-n}$ applies when $m \neq n$.

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

(D) If $m=n$, the external division point is the midpoint of AB.

Question 6. If R divides PQ internally in ratio m:n, which is FALSE about vectors $\vec{PR}$ and $\vec{RQ}$?

(A) $\vec{PR}$ and $\vec{RQ}$ are in the same direction.

(B) $|\vec{PR}| : |\vec{RQ}| = m:n$.

(C) $\vec{PR} = \frac{m}{n}\vec{RQ}$.

(D) $\vec{PR} = \frac{m}{m+n}\vec{PQ}$ and $\vec{RQ} = \frac{n}{m+n}\vec{PQ}$.

Question 7. If A, B, C, D are vertices of a tetrahedron, which is NOT the formula for the position vector of its centroid G?

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

(B) The point dividing the line joining vertex A to the centroid of face BCD in ratio 3:1.

(C) Average of the position vectors of the four vertices.

(D) Intersection of the medians of the faces.

Question 8. If $\vec{r} = (1-\lambda)\vec{a} + \lambda\vec{b}$ is the equation of the line AB, which point on AB does NOT correspond to a value of $\lambda$ in the range $[0,1]$?

(A) Point A ($\lambda=0$)

(B) Point B ($\lambda=1$)

(C) The midpoint of AB ($\lambda=1/2$)

(D) A point R dividing AB externally ($\lambda = 2$, for example)

Question 9. Which ratio $m:n$ for dividing a line segment AB (A to B) by a point R corresponds to R being on the line segment AB?

(A) 2:1 (Internal)

(B) -2:1 (This corresponds to external division 2:1)

(C) 3:(-2) (This corresponds to external division 3:2)

(D) 1:1 (Internal)

Question 10. Which property of points on a line is NOT directly derived or related to the section formula?

(A) Collinearity of three points.

(B) Position of the midpoint.

(C) Position of the centroid of a triangle formed by one point and two other points (unless they are the same point).

(D) Dividing a segment in a given ratio.

Question 1. Which equation does NOT represent a coordinate plane in 3D space?

(A) $x = 0$

(B) $y = 0$

(C) $z = 0$

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

Question 2. Which formula for distance in 3D is INCORRECT?

(A) Distance between $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$.

(B) Distance of $(x, y, z)$ from origin is $\sqrt{x^2+y^2+z^2}$.

(C) Distance of $(x, y, z)$ from xy-plane is $|z|$.

(D) Distance between $\vec{a}$ and $\vec{b}$ is $|\vec{a}| + |\vec{b}|$.

Question 3. Which set of three values can NOT be direction cosines of a line?

(A) $(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}})$

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

(C) $(1/2, 1/2, 1/2)$

(D) $(1/3, 2/3, 2/3)$

Question 4. Which is NOT a correct form of the vector equation of a line?

(A) $\vec{r} = \vec{a} + \lambda \vec{b}$ (point $\vec{a}$, parallel vector $\vec{b}$)

(B) $\vec{r} = (1-\lambda)\vec{a} + \lambda\vec{b}$ (passes through $\vec{a}$ and $\vec{b}$)

(C) $\vec{r} \cdot \vec{n} = d$ (equation of a plane)

(D) $(\vec{r} - \vec{a}) \times \vec{n} = \vec{0}$ (This is the equation of a line)

Question 5. Which is NOT a correct form of the Cartesian equation of a line?

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

(B) $x = x_1 + \lambda a, y = y_1 + \lambda b, z = z_1 + \lambda c$ (Parametric form)

(C) $\frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1}$ (Two-point form)

(D) $a(x-x_1) + b(y-y_1) + c(z-z_1) = 0$ (Equation of a plane perpendicular to direction $(a,b,c)$ at $(x_1,y_1,z_1)$)

Question 6. Which statement about direction ratios $(a, b, c)$ of a line is FALSE?

(A) They are proportional to the direction cosines $(\ell, m, n)$.

(B) $(0, 0, 0)$ can be the direction ratios of a line.

(C) If $(a, b, c)$ are direction ratios, then $(ka, kb, kc)$ are also direction ratios for any non-zero $k$.

(D) They represent the components of a vector parallel to the line.

Question 7. Which point does NOT lie on the line $\frac{x-2}{1} = \frac{y+1}{2} = \frac{z-3}{-1}$?

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

(B) $(3, 1, 2)$ (for $\lambda=1$)

(C) $(1, -3, 4)$ (for $\lambda=-1$)

(D) $(0, -5, 5)$ (for $\lambda=-2$)

Question 8. Which statement is NOT true about the vector equation of a line $\vec{r} = \vec{a} + \lambda \vec{b}$?

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

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

(C) Varying $\lambda$ generates different points on the line.

(D) If $\vec{b}$ is a unit vector, $\lambda$ represents the distance from point $\vec{a}$.

Question 9. Which conversion between vector and Cartesian forms of a line is INCORRECT?

(A) $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ in vector form corresponds to $(x,y,z)$ in Cartesian form.

(B) $\vec{r} = (x_1\hat{i} + y_1\hat{j} + z_1\hat{k}) + \lambda (a\hat{i} + b\hat{j} + c\hat{k})$ converts to $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$.

(C) $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} = \lambda$ converts to $x=x_1+\lambda a, y=y_1+\lambda b, z=z_1+\lambda c$.

(D) The Cartesian equation always involves squares of variables.

Question 10. Which pair of direction cosines corresponds to a line parallel to a coordinate axis?

(A) $(1, 0, 0)$ (x-axis)

(B) $(0, 1, 0)$ (y-axis)

(C) $(0, 0, 1)$ (z-axis)

(D) $(1/\sqrt{2}, 1/\sqrt{2}, 0)$ (line in xy-plane at 45 degrees)

Question 1. Which is NOT a correct form of the equation of a plane?

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

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

(C) $Ax + By + Cz + D = 0$ (General Cartesian form)

(D) $\vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c}$ (Vector form of a line)

Question 2. Which vector is NOT necessarily a normal vector to the plane $Ax + By + Cz + D = 0$?

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

(B) $k(A\hat{i} + B\hat{j} + C\hat{k})$ for any non-zero scalar $k$.

(C) A vector parallel to the plane.

(D) The vector whose components are the coefficients of x, y, and z.

Question 3. Which is NOT the equation of a plane passing through a point $\vec{a}$ and perpendicular to a vector $\vec{n}$?

(A) $(\vec{r} - \vec{a}) \cdot \vec{n} = 0$

(B) $\vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n}$

(C) $\vec{r} \cdot \vec{n} = d$ where $d = \vec{a} \cdot \vec{n}$

(D) $(\vec{r} - \vec{a}) \times \vec{n} = \vec{0}$ (This is the equation of a line)

Question 4. Which statement about the intercept form of the plane equation $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ is FALSE?

(A) $a$ is the x-intercept, $b$ is the y-intercept, and $c$ is the z-intercept.

(B) The plane passes through the points $(a,0,0), (0,b,0), (0,0,c)$.

(C) This form is valid for any plane (including those passing through the origin or parallel to axes).

(D) The intercepts $a, b, c$ must be non-zero.

Question 5. Which is NOT the equation of a plane passing through the origin?

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

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

(C) $x+y+z=1$

(D) The plane perpendicular to $\vec{n}$ at the origin.

Question 6. The equation of any plane passing through the intersection of planes $P_1=0$ and $P_2=0$ is $P_1 + \lambda P_2 = 0$. Which is NOT true about this family of planes?

(A) It represents all planes containing the line of intersection of $P_1$ and $P_2$.

(B) For different values of $\lambda$, we get different planes (except for the special case of $P_2$ itself). For $\lambda=0$, we get $P_1$.

(C) If $P_1$ and $P_2$ are parallel, $P_1 + \lambda P_2 = 0$ still represents a family of parallel planes.

(D) If $P_1$ and $P_2$ are identical, $P_1 + \lambda P_2 = 0$ represents any plane in space.

Question 7. Which statement about the distance of a plane from the origin is INCORRECT?

(A) For the plane $Ax+By+Cz+D=0$, the distance is $\frac{|D|}{\sqrt{A^2+B^2+C^2}}$.

(B) For the plane $\vec{r} \cdot \hat{n} = d$, the distance is $|d|$.

(C) For the plane $\vec{r} \cdot \vec{n} = k$, the distance is $\frac{|k|}{|\vec{n}|}$.

(D) The distance is always positive.

Question 8. Which set of points does NOT uniquely determine a plane?

(A) Three non-collinear points.

(B) A point and a normal vector.

(C) Two distinct points.

(D) A line and a point not on the line.

Question 9. Which conversion between vector and Cartesian forms of a plane is INCORRECT?

(A) $\vec{r} \cdot (A\hat{i} + B\hat{j} + C\hat{k}) = D'$ corresponds to $Ax+By+Cz=D'$.

(B) $Ax+By+Cz+D=0$ corresponds to $\vec{r} \cdot (A\hat{i} + B\hat{j} + C\hat{k}) = -D$.

(C) $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ corresponds to $\vec{r} \cdot (\frac{1}{a}\hat{i} + \frac{1}{b}\hat{j} + \frac{1}{c}\hat{k}) = 1$.

(D) $(\vec{r} - \vec{a}) \cdot \vec{n} = 0$ corresponds to $\vec{r} = \vec{a} + \lambda \vec{n}$.

Question 10. Which property is FALSE for two parallel planes $P_1$ and $P_2$?

(A) Their normal vectors are proportional.

(B) The distance between any point on $P_1$ and the plane $P_2$ is constant.

(C) They intersect in a line.

(D) If they are distinct, they have no common points.

Question 1. Which is NOT a correct formula for the cosine of the acute angle $\theta$ between two lines with direction vectors $\vec{b}_1$ and $\vec{b}_2$?

(A) $\frac{|\vec{b}_1 \cdot \vec{b}_2|}{|\vec{b}_1||\vec{b}_2|}$

(B) $\frac{\vec{b}_1 \cdot \vec{b}_2}{|\vec{b}_1||\vec{b}_2|}$ (This gives the cosine, which could be negative for obtuse angle)

(C) $\frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$ (using direction ratios)

(D) $|\cos\theta|$ where $\theta$ is the angle between $\vec{b}_1$ and $\vec{b}_2$

Question 2. Which is NOT a correct condition for two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ to be perpendicular?

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

(B) Their direction vectors are perpendicular.

(C) $a_1a_2 + b_1b_2 + c_1c_2 = 0$.

(D) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

Question 3. Which is NOT a correct formula for the cosine of the angle $\theta$ between two planes with normal vectors $\vec{n}_1$ and $\vec{n}_2$?

(A) $\frac{|\vec{n}_1 \cdot \vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$

(B) $\frac{|A_1A_2 + B_1B_2 + C_1C_2|}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}$ (using coefficients of $A_ix+B_iy+C_iz+D_i=0$)

(C) $\frac{\vec{n}_1 \cdot \vec{n}_2}{|\vec{n}_1||\vec{n}_2|}$ (This gives the cosine, which could be negative for obtuse angle)

(D) $\sin(90^\circ - \phi)$ where $\phi$ is the angle between the planes.

Question 4. Which is NOT a correct condition for two planes with normal vectors $\vec{n}_1$ and $\vec{n}_2$ to be parallel?

(A) $\vec{n}_1 = k\vec{n}_2$ for some non-zero scalar $k$.

(B) Their normal vectors are collinear.

(C) The angle between the planes is $0^\circ$ or $180^\circ$.

(D) $\vec{n}_1 \cdot \vec{n}_2 = 0$

Question 5. Which is NOT a correct formula for the sine of the angle $\phi$ between a line with direction vector $\vec{b}$ and a plane with normal vector $\vec{n}$?

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

(B) $\cos\theta$ where $\theta$ is the angle between the line and the normal to the plane.

(C) $\frac{|aA + bB + cC|}{\sqrt{a^2+b^2+c^2}\sqrt{A^2+B^2+C^2}}$ (using direction ratios and normal coefficients)

(D) $\frac{|\vec{b} \times \vec{n}|}{|\vec{b}||\vec{n}|}$

Question 6. If a line with direction vector $\vec{b}$ is parallel to a plane with normal vector $\vec{n}$, which is NOT true?

(A) $\vec{b}$ is perpendicular to $\vec{n}$.

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

(C) The angle between the line and the plane is $0^\circ$ or $180^\circ$.

(D) $\vec{b}$ is parallel to $\vec{n}$.

Question 7. If a line with direction vector $\vec{b}$ is perpendicular to a plane with normal vector $\vec{n}$, which is FALSE?

(A) $\vec{b}$ is parallel to $\vec{n}$.

(B) $\vec{b} = k\vec{n}$ for some non-zero scalar $k$.

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

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

Question 8. Which dot product condition does NOT correspond to a $90^\circ$ angle between the vectors $\vec{u}$ and $\vec{v}$?

(A) $\vec{u} \cdot \vec{v} = 0$

(B) $|\vec{u} + \vec{v}|^2 = |\vec{u}|^2 + |\vec{v}|^2$

(C) $|\vec{u} - \vec{v}|^2 = |\vec{u}|^2 + |\vec{v}|^2$

(D) $\vec{u} \cdot \vec{v} = |\vec{u}||\vec{v}|$

Question 9. Which pair of lines are NOT parallel based on their direction ratios?

(A) $(1, 2, 3)$ and $(2, 4, 6)$

(B) $(-1, -2, -3)$ and $(1, 2, 3)$

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

(D) $(2, 3, -1)$ and $(4, 6, -2)$

Question 10. Which pair of planes are NOT perpendicular based on their normal vectors?

(A) $x+y+z=1$ and $x-y=0$ (Normals: $(1,1,1)$ and $(1,-1,0)$)

(B) $2x+y-3z=0$ and $x+y+z=5$ (Normals: $(2,1,-3)$ and $(1,1,1)$)

(C) $z=0$ and $x=0$ (Normals: $(0,0,1)$ and $(1,0,0)$)

(D) $x=1$ and $y=2$ (Normals: $(1,0,0)$ and $(0,1,0)$)

Question 1. Which is NOT the correct formula for the shortest distance between two skew lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$?

(A) $\frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}$

(B) The length of the common perpendicular between the lines.

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

(D) $\frac{|(\vec{a}_2 - \vec{a}_1) \times (\vec{b}_1 \cdot \vec{b}_2)|}{|\vec{b}_1 \cdot \vec{b}_2|}$

Question 2. Which is NOT the correct formula for the shortest distance between two parallel lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}$?

(A) $\frac{|(\vec{a}_2 - \vec{a}_1) \times \vec{b}|}{|\vec{b}|}$

(B) $\frac{|\vec{b} \times (\vec{a}_1 - \vec{a}_2)|}{|\vec{b}|}$

(C) The distance of point $\vec{a}_1$ from the line $\vec{r} = \vec{a}_2 + \mu \vec{b}$.

(D) $\frac{|(\vec{a}_2 - \vec{a}_1) \cdot \vec{b}|}{|\vec{b}|}$

Question 3. Which is NOT the correct formula for the distance of a point P$(x_1, y_1, z_1)$ from the plane $Ax + By + Cz + D = 0$?

(A) $\frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2+B^2+C^2}}$

(B) $\frac{|\vec{p} \cdot \vec{n} + D|}{|\vec{n}|}$ where $\vec{p} = x_1\hat{i} + y_1\hat{j} + z_1\hat{k}$ and $\vec{n} = A\hat{i} + B\hat{j} + C\hat{k}$.

(C) $\frac{|Ax_1 + By_1 + Cz_1 + D|}{A^2+B^2+C^2}$

(D) The length of the perpendicular from the point to the plane.

Question 4. Which property is FALSE about skew lines?

(A) They are non-parallel.

(B) They do not intersect.

(C) They are coplanar.

(D) They exist only in 3D space.

Question 5. If the shortest distance between two lines is 0, which is NOT necessarily true?

(A) The lines are intersecting.

(B) The lines are coincident.

(C) The lines are coplanar.

(D) The lines are perpendicular.

Question 6. Which formula for the distance of a point P from a line L (passing through A with direction $\vec{b}$) is INCORRECT?

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

(D) $\vec{AP} \cdot \frac{\vec{b}}{|\vec{b}|}$

Question 7. Which statement about the shortest distance between two parallel planes is FALSE?

(A) The distance is zero if the planes are identical.

(B) The formula $\frac{|D_2 - D_1|}{\sqrt{A^2+B^2+C^2}}$ applies to planes $Ax+By+Cz+D_1=0$ and $Ax+By+Cz+D_2=0$.

(C) The distance is measured along the common normal to the planes.

(D) The distance depends on the specific points chosen on each plane for measurement.

Question 8. Which pair of planes does NOT have a zero distance between them?

(A) $x=1$ and $x=1$

(B) $x+y+z=1$ and $2x+2y+2z=2$

(C) $x+y+z=1$ and $x+y+z=5$

(D) A plane and itself.

Question 9. Which is NOT the distance of the origin $(0,0,0)$ from a plane?

(A) $\frac{|D|}{\sqrt{A^2+B^2+C^2}}$ for $Ax+By+Cz+D=0$.

(B) $|d|$ for $\vec{r} \cdot \hat{n} = d$.

(C) $\frac{|\vec{a} \cdot \vec{n}|}{|\vec{n}|}$ for a plane passing through point $\vec{a}$ with normal $\vec{n}$.

(D) $\sqrt{x_1^2 + y_1^2 + z_1^2}$ where $(x_1, y_1, z_1)$ is the foot of the perpendicular from the origin to the plane.

Question 10. The shortest distance between two lines is measured along which type of line segment?

(A) A line segment connecting any two points on the lines.

(B) The shortest line segment connecting a point on one line to a point on the other.

(C) A line segment perpendicular to both lines.

(D) The common perpendicular (if it exists and is finite).