CBSE Class 12 Mathematics Vectors Algebra MCQs Set 06

Question. Let \( \vec{a} \) and \( \vec{b} \) be two unit vectors and \( \theta \) is the angle between them. Then \( \vec{a} + \vec{b} \) is unit vector if \( \theta \) is
(a) \( \frac{\pi}{4} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{2} \)
(d) \( \frac{2\pi}{3} \)
Answer: (d) \( \frac{2\pi}{3} \)

Question. The value of \( \hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}) \) is
(a) 0
(b) -1
(c) 1
(d) 3
Answer: (c) 1

Question. If \( \theta \) is the angle between any two vectors \( \vec{a} \) and \( \vec{b} \) then \( |\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}| \), where \( \theta \) is equal to
(a) 0
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{2} \)
(d) \( \pi \)
Answer: (d) \( \pi \)

Question. The magnitude of the vector \( 6\hat{i} + 2\hat{j} + 3\hat{k} \) is 
(a) 5
(b) 7
(c) 12
(d) 1
Answer: (b) 7

Question. The vector of the direction of the vector \( \hat{i} - 2\hat{j} + 2\hat{k} \) that has magnitude 9 is
(a) \( \hat{i} - 2\hat{j} + 2\hat{k} \)
(b) \( \frac{\hat{i} - 2\hat{j} + 2\hat{k}}{3} \)
(c) \( 3(\hat{i} - 2\hat{j} + 2\hat{k}) \)
(d) \( 9(\hat{i} - 2\hat{j} + 2\hat{k}) \)
Answer: (c) \( 3(\hat{i} - 2\hat{j} + 2\hat{k}) \)

Question. The position vector of the point which divides the join of point \( 2\vec{a} - 3\vec{b} \) and \( \vec{a} + \vec{b} \) in the ratio 3 : 1 is
(a) \( \frac{3\vec{a} - 2\vec{b}}{2} \)
(b) \( \frac{7\vec{a} - 8\vec{b}}{4} \)
(c) \( \frac{3\vec{a}}{4} \)
(d) \( \frac{5\vec{a}}{4} \)
Answer: (d) \( \frac{5\vec{a}}{4} \)

Question. The vector having initial and terminal points as (2, 5, 0) and (–3, 7, 4) respectively is
(a) \( -\hat{i} + 12\hat{j} + 4\hat{k} \)
(b) \( 5\hat{i} + 2\hat{j} - 4\hat{k} \)
(c) \( -5\hat{i} + 2\hat{j} + 4\hat{k} \)
(d) \( \hat{i} + \hat{j} + \hat{k} \)
Answer: (c) \( -5\hat{i} + 2\hat{j} + 4\hat{k} \)

Question. The angle between two vectors \( \vec{a} \) and \( \vec{b} \) with magnitudes \( \sqrt{3} \) and 4 respectively and \( \vec{a} \cdot \vec{b} = 2\sqrt{3} \) is 
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{2} \)
(d) \( \frac{5\pi}{2} \)
Answer: (b) \( \frac{\pi}{3} \)

Question. Find the value of \( \lambda \) such that the vectors \( \vec{a} = 2\hat{i} + \lambda\hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} + 3\hat{k} \) are orthogonal
(a) 0
(b) 1
(c) \( \frac{3}{2} \)
(d) \( -\frac{5}{2} \)
Answer: (d) \( -\frac{5}{2} \)

Question. The value of \( \lambda \) for which the vectors \( 3\hat{i} - 6\hat{j} + \hat{k} \) and \( 2\hat{i} - 4\hat{j} + \lambda\hat{k} \) are parallel is
(a) \( \frac{2}{3} \)
(b) \( \frac{3}{2} \)
(c) \( \frac{5}{2} \)
(d) \( \frac{2}{5} \)
Answer: (a) \( \frac{2}{3} \)

Question. The vector from origin to the points A and B are \( \vec{a} = 2\hat{i} - 3\hat{j} + 2\hat{k} \) and \( \vec{b} = 2\hat{i} + 3\hat{j} + \hat{k} \), respectively then the area of triangle OAB is 
(a) 340
(b) \( \sqrt{25} \)
(c) \( \sqrt{229} \)
(d) \( \frac{1}{2}\sqrt{229} \)
Answer: (d) \( \frac{1}{2}\sqrt{229} \)

Question. For any vector \( \vec{a} \), the value of \( (\vec{a} \times \hat{i})^2 + (\vec{a} \times \hat{j})^2 + (\vec{a} \times \hat{k})^2 \) is equal to 
(a) \( \vec{a}^2 \)
(b) \( 3\vec{a}^2 \)
(c) \( 4\vec{a}^2 \)
(d) \( 2\vec{a}^2 \)
Answer: (d) \( 2\vec{a}^2 \)

Question. If \( |\vec{a}| = 10, |\vec{b}| = 2 \) and \( \vec{a} \cdot \vec{b} = 12 \), then value of \( |\vec{a} \times \vec{b}| \) is
(a) 5
(b) 10
(c) 14
(d) 16
Answer: (d) 16

Question. The vector \( \lambda\hat{i} + \hat{j} + 2\hat{k}, \hat{i} + \lambda\hat{j} - \hat{k} \) and \( 2\hat{i} - \hat{j} + \lambda\hat{k} \) are coplanar if
(a) \( \lambda = -2 \)
(b) \( \lambda = 0 \)
(c) \( \lambda = 1 \)
(d) \( \lambda = -1 \)
Answer: (a) \( \lambda = -2 \)

Question. If \( \vec{a}, \vec{b}, \vec{c} \) are unit vectors such that \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \), then the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \) is
(a) 1
(b) 3
(c) \( -\frac{3}{2} \)
(d) None of the options
Answer: (c) \( -\frac{3}{2} \)

Question. Projection vector of \( \vec{a} \) on \( \vec{b} \) is
(a) \( \left( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \right) \vec{b} \)
(b) \( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \)
(c) \( \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|} \)
(d) \( \left( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \right) \hat{b} \)
Answer: (a) \( \left( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \right) \vec{b} \)

Question. If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are three vectors such that \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \) and \( |\vec{a}| = 2, |\vec{b}| = 3, |\vec{c}| = 5 \), then value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \) is
(a) 0
(b) 1
(c) -19
(d) 38
Answer: (c) -19

Question. If \( |\vec{a}| = 4 \) and \( -3 \le \lambda \le 2 \), then the range of \( |\lambda \vec{a}| \) is 
(a) [0, 8]
(b) [-12, 8]
(c) [0, 12]
(d) [8, 12]
Answer: (c) [0, 12]

Question. The number of vectors of unit length perpendicular to the vectors \( \vec{a} = 2\hat{i} + \hat{j} + 2\hat{k} \) and \( \vec{b} = \hat{j} + \hat{k} \) is
(a) one
(b) two
(c) three
(d) infinite
Answer: (b) two

Question. The position vector of the point which divides the join of points with position vectors \( \vec{a} + \vec{b} \) and \( 2\vec{a} - \vec{b} \) in the ratio 1 : 2 is 
(a) \( \frac{3\vec{a} + 2\vec{b}}{3} \)
(b) \( \vec{a} \)
(c) \( \frac{5\vec{a} - \vec{b}}{3} \)
(d) \( \frac{4\vec{a} + \vec{b}}{3} \)
Answer: (d) \( \frac{4\vec{a} + \vec{b}}{3} \)

Assertion-Reason Questions 

The following questions consist of two statements—Assertion(A) and Reason(R). Answer these questions selecting the appropriate option given below:
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (A) : Direction cosines of vector \( \vec{a} = \hat{i} + \hat{j} - 2\hat{k} \) are \( \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{-2}{\sqrt{6}} \).
Reason (R) : If vector \( \vec{r} = a\hat{i} + b\hat{j} + c\hat{k} \) then its direction ratios are \( \frac{a}{|\vec{r}|}, \frac{b}{|\vec{r}|}, \frac{c}{|\vec{r}|} \), where \( |\vec{r}| = \sqrt{a^2 + b^2 + c^2} \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.

Question. Assertion (A) : If \( (\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = 0 \), then \( \vec{a} \) and \( \vec{b} \) are perpendicular.
Reason (R) : The projection of \( \hat{i} + 3\hat{j} + \hat{k} \) on \( 2\hat{i} - 3\hat{j} + 6\hat{k} \) is \( -\frac{1}{7} \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both A and R are true and R is not the correct explanation for A.

Question. Assertion (A) : If \( \vec{a} = 3\hat{i} - \hat{j} + 2\hat{k}, \vec{b} = 2\hat{i} + 3\hat{j} + 3\hat{k} \) then \( \vec{a} \cdot \vec{b} = 9 \).
Reason (R) : If \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) then its magnitude \( |\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both A and R are true and R is not the correct explanation for A.

Question. Assertion (A) : Cosine of the angle between the two vectors \( 2\hat{i} + 2\hat{j} - \hat{k} \) and \( 6\hat{i} - 3\hat{j} + 2\hat{k} \) is \( \frac{16}{21} \).
Reason (R) : Cosine of the angle between two vectors \( \vec{a} \) and \( \vec{b} \) is given by \( \cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (d) A is false but R is true.

Question. Assertion (A) : If \( |\vec{a} \times \vec{b}| = 1 \) and \( |\vec{a} \cdot \vec{b}| = \sqrt{3} \) then the angle between \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{6} \).
Reason (R) : \( |\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta \) and \( |\vec{a} \cdot \vec{b}| = |\vec{a}||\vec{b}|\cos\theta \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.