Check the below NCERT MCQ Questions for Class 12 Maths Chapter 10 Vector Algebra with Answers Pdf free download. MCQ Questions for Class 12 Maths with Answers were prepared based on the latest exam pattern. We have provided Vector Algebra Class 12 Maths MCQs Questions with Answers to help students understand the concept very well.

## Vector Algebra Class 12 MCQs Questions with Answers

Vector MCQ Chapter 10 Class 12 Question 1.
The position vector of the point (1, 0, 2) is
(a) $$\vec{i}$$ +$$\vec{j}$$ + 2$$\vec{k}$$
(b) $$\vec{i}$$ + 2$$\vec{j}$$
(c) $$\vec{2}$$ + 3$$\vec{k}$$
(d) $$\vec{i}$$ + 2$$\vec{K}$$

Answer: (d) $$\vec{i}$$ + 2$$\vec{K}$$

Vector MCQ Questions Chapter 10 Class 12 Question 2.
The modulus of 7$$\vec{i}$$ – 2$$\vec{J}$$ + $$\vec{K}$$
(a) $$\sqrt{10}$$
(b) $$\sqrt{55}$$
(c) 3$$\sqrt{6}$$
(d) 6

Answer: (c) 3$$\sqrt{6}$$

MCQ On Vectors Chapter 10 Class 12 Question 3.
If O be the origin and $$\vec{OP}$$ = 2$$\hat{i}$$ + 3$$\hat{j}$$ – 4$$\hat{k}$$ and $$\vec{OQ}$$ = 5$$\hat{i}$$ + 4$$\hat{j}$$ -3$$\hat{k}$$, then $$\vec{PQ}$$ is equal to
(a) 7$$\hat{i}$$ + 7$$\hat{j}$$ – 7$$\hat{k}$$
(b) -3$$\hat{i}$$ + $$\hat{j}$$ – $$\hat{k}$$
(c) -7$$\hat{i}$$ – 7$$\hat{j}$$ + 7$$\hat{k}$$
(d) 3$$\hat{i}$$ + $$\hat{j}$$ + $$\hat{k}$$

Answer: (d) 3$$\hat{i}$$ + $$\hat{j}$$ + $$\hat{k}$$

MCQs On Vectors Chapter 10 Class 12 Question 4.
The scalar product of 5$$\hat{i}$$ + $$\hat{j}$$ – 3$$\hat{k}$$ and 3$$\hat{i}$$ – 4$$\hat{j}$$ + 7$$\hat{k}$$ is
(a) 10
(b) -10
(c) 15
(d) -15

MCQ On Vector Chapter 10 Class 12 Question 5.
If $$\vec{a}$$.$$\vec{b}$$ = 0, then
(a) a ⊥ b
(b) $$\vec{a}$$ || $$\vec{b}$$
(c) $$\vec{a}$$ + $$\vec{b}$$ = 0
(d) $$\vec{a}$$ – $$\vec{b}$$ = 0

Vectors MCQs Chapter 10 Class 12 Question 6.
$$\vec{i}$$ – $$\vec{j}$$ =
(a) 0
(b) 1
(c) $$\vec{k}$$
(d) –$$\vec{k}$$

MCQ On Vectors Class 12 Chapter 10 Question 7.
$$\vec{k}$$ × $$\vec{j}$$ =
(a) 0
(b) 1
(c) $$\vec{i}$$
(d) –$$\vec{i}$$

Answer: (d) –$$\vec{i}$$

Vectors MCQ Chapter 10 Class 12 Question 8.
$$\vec{a}$$. $$\vec{a}$$ =
(a) 0
(b) 1
(c) |$$\vec{a}$$|²
(d) |$$\vec{a}$$|

Answer: (c) |$$\vec{a}$$|²

Vector MCQ Questions Class 12 Chapter 10 Question 9.
The projection of the vector 2$$\hat{i}$$ – $$\hat{j}$$ + $$\hat{k}$$ on the vector $$\hat{i}$$ – 2$$\hat{j}$$ + $$\hat{k}$$ is
(a) $$\frac{4}{√6}$$
(b) $$\frac{5}{√6}$$
(c) $$\frac{4}{√3}$$
(d) $$\frac{7}{√6}$$

Answer: (b) $$\frac{5}{√6}$$

Vectors Are MCQ Chapter 10 Class 12 Question 10.
If $$\vec{a}$$ = $$\vec{i}$$ – $$\vec{j}$$ + 2$$\vec{k}$$ and b = 3$$\vec{i}$$ + 2$$\vec{j}$$ – $$\vec{k}$$ then the value of ($$\vec{a}$$ + 3$$\vec{b}$$)(2$$\vec{a}$$ – $$\vec{b}$$)=.
(a) 15
(b) -15
(c) 18
(d) -18

MCQ On Vector Algebra Chapter 10 Class 12 Question 11.
If |$$\vec{a}$$|= $$\sqrt{26}$$, |b| = 7 and |$$\vec{a}$$ × $$\vec{b}$$| = 35, then $$\vec{a}$$.$$\vec{b}$$ =
(a) 8
(b) 7
(c) 9
(d) 12

Vector Algebra MCQ Chapter 10 Class 12 Question 12.
If $$\vec{a}$$ = 2$$\vec{i}$$ – 3$$\vec{j}$$ + 4$$\vec{k}$$ and $$\vec{b}$$ = $$\vec{i}$$ + 2$$\vec{j}$$ + $$\vec{k}$$ then $$\vec{a}$$ + $$\vec{b}$$ =
(a) $$\vec{i}$$ + $$\vec{j}$$ + 3$$\vec{k}$$
(b) 3$$\vec{i}$$ – $$\vec{j}$$ + 5$$\vec{k}$$
(c) $$\vec{i}$$ – $$\vec{j}$$ – 3$$\vec{k}$$
(d) 2$$\vec{i}$$ + $$\vec{j}$$ + $$\vec{k}$$

Answer: (b) 3$$\vec{i}$$ – $$\vec{j}$$ + 5$$\vec{k}$$

Vector MCQs Chapter 10 Class 12 Question 13.
If $$\vec{a}$$ = $$\vec{i}$$ + 2$$\vec{j}$$ + 3$$\vec{k}$$ and $$\vec{b}$$ = 3$$\vec{i}$$ + 2$$\vec{j}$$ + $$\vec{k}$$, then cos θ =
(a) $$\frac{6}{7}$$
(b) $$\frac{5}{7}$$
(c) $$\frac{4}{7}$$
(d) $$\frac{1}{2}$$

Answer: (b) $$\frac{5}{7}$$

Vectors MCQs With Solutions Chapter 10 Class 12 Question 14.
If |$$\vec{a}$$ + $$\vec{b}$$| = |$$\vec{a}$$ – $$\vec{b}$$|, then
(a) $$\vec{a}$$ || $$\vec{a}$$
(b) $$\vec{a}$$ ⊥ $$\vec{b}$$
(c) |$$\vec{a}$$| = |$$\vec{b}$$|
(d) None of these

Answer: (b) $$\vec{a}$$ ⊥ $$\vec{b}$$

MCQ Questions On Vectors Chapter 10 Class 12 Question 15.
The projection of the vector 2$$\hat{i}$$ + 3$$\hat{j}$$ – 6$$\hat{k}$$ on the line joining the points (3, 4, 2) and (5, 6,3) is
(a) $$\frac{2}{3}$$
(b) $$\frac{4}{3}$$
(c) –$$\frac{4}{3}$$
(d) $$\frac{5}{3}$$

Answer: (b) $$\frac{4}{3}$$

Question 16.
If |$$\vec{a}$$ × $$\vec{b}$$| – |$$\vec{a}$$.$$\vec{b}$$|, then the angle between $$\vec{a}$$ and $$\vec{b}$$, is
(a) 0
(b) $$\frac{π}{2}$$
(c) $$\frac{π}{4}$$
(d) π

Answer: (c) $$\frac{π}{4}$$

Question 17.
The angle between two vector $$\vec{a}$$ and $$\vec{b}$$ with magnitude √3 and 4, respectively and $$\vec{a}$$.$$\vec{b}$$ = 2√3 is
(a) $$\frac{π}{6}$$
(b) $$\frac{π}{3}$$
(c) $$\frac{π}{2}$$
(d) $$\frac{5π}{2}$$

Answer: (b) $$\frac{π}{3}$$

Question 18.
Unit vector perpendicular to each of the vector 3$$\hat{i}$$ + $$\hat{j}$$ + 2$$\hat{k}$$ and 2$$\hat{i}$$ – 2$$\hat{j}$$ + 4$$\hat{k}$$ is
(a) $$\frac{\hat{i}+\hat{j}+\hat{k}}{√3}$$
(b) $$\frac{\hat{i}-\hat{j}+\hat{k}}{√3}$$
(c) $$\frac{\hat{i}-\hat{j}-\hat{k}}{√3}$$
(d) $$\frac{\hat{i}+\hat{j}-\hat{k}}{√3}$$

Answer: (c) $$\frac{\hat{i}-\hat{j}-\hat{k}}{√3}$$

Question 19.
If $$\vec{a}$$ = 2$$\vec{i}$$ – 5$$\vec{j}$$ + k and $$\vec{b}$$ = 4$$\vec{i}$$ + 2$$\vec{j}$$ + $$\vec{k}$$ then $$\vec{a}$$.$$\vec{b}$$ =
(a) 0
(b) -1
(c) 1
(d) 2

Question 20.
If 2$$\vec{i}$$ + $$\vec{j}$$ + $$\vec{k}$$, 6$$\vec{i}$$ – $$\vec{j}$$ + 2$$\vec{k}$$ and 14$$\vec{i}$$ – 5$$\vec{j}$$ + 4$$\vec{k}$$ be the position vector of the points A, B and C respectively, then
(a) The A, B and C are collinear
(b) A, B and C are not colinear
(c) $$\vec{AB}$$ ⊥ $$\vec{BC}$$
(d) None of these

Answer: (a) The A, B and C are collinear

Question 21.
According to the associative lass of addition of addition of s ector
($$\vec{a}$$ + …….) + $$\vec{c}$$ = …… + ($$\vec{b}$$ + $$\vec{c}$$)
(a) $$\vec{b}$$, $$\vec{a}$$
(b) $$\vec{a}$$, $$\vec{b}$$
(c) $$\vec{a}$$, 0
(d) $$\vec{b}$$, 0

Answer: (a) $$\vec{b}$$, $$\vec{a}$$

Question 22.
Which one of the following can be written for ($$\vec{a}$$ – $$\vec{b}$$) × ($$\vec{a}$$ + $$\vec{b}$$)
(a) $$\vec{a}$$ × $$\vec{b}$$
(b) 2$$\vec{a}$$ × $$\vec{b}$$
(c) $$\vec{a}$$² – $$\vec{b}$$
(d) 2$$\vec{b}$$ × $$\vec{b}$$

Answer: (b) 2$$\vec{a}$$ × $$\vec{b}$$

Question 23.
The points with position vectors (2. 6), (1, 2) and (a, 10) are collinear if the of a is
(a) -8
(b) 4
(c) 3
(d) 12

Question 24.
|$$\vec{a}$$ + $$\vec{b}$$| = |$$\vec{a}$$ – $$\vec{b}$$| then the angle between $$\vec{a}$$ and $$\vec{b}$$
(a) $$\frac{π}{2}$$
(b) 0
(c) $$\frac{π}{4}$$
(d) $$\frac{π}{6}$$

Answer: (a) $$\frac{π}{2}$$

Question 25.
|$$\vec{a}$$ × $$\vec{b}$$| = |$$\vec{a}$$.$$\vec{b}$$| then the angle between $$\vec{a}$$ and $$\vec{b}$$
(a) 0
(b) $$\frac{π}{2}$$
(c) $$\frac{π}{4}$$
(d) π

Question 26.
If ABCDEF is a regular hexagon then $$\vec{AB}$$ + $$\vec{EB}$$ + $$\vec{FC}$$ equals
(a) zero
(b) 2$$\vec{AB}$$
(c) 4$$\vec{AB}$$
(d) 3$$\vec{AB}$$

Answer: (d) 3$$\vec{AB}$$

Question 27.
Which one of the following is the modulus of x$$\hat{i}$$ + y$$\hat{j}$$ + z$$\hat{k}$$?
(a) $$\sqrt{x^2+y^2+z^2}$$
(b) $$\frac{1}{\sqrt{x^2+y^2+z^2}}$$
(c) x² + y² + z²
(d) none of these

Answer: (a) $$\sqrt{x^2+y^2+z^2}$$

Question 28.
If C is the mid point of AB and P is any point outside AB then,
(a) $$\vec{PA}$$ + $$\vec{PB}$$ = 2$$\vec{PC}$$
(b) $$\vec{PA}$$ + $$\vec{PB}$$ = $$\vec{PC}$$
(c) $$\vec{PA}$$ + $$\vec{PB}$$ = 2$$\vec{PC}$$ = 0
(d) None of these

Answer: (a) $$\vec{PA}$$ + $$\vec{PB}$$ = 2$$\vec{PC}$$

Question 29.
If $$\vec{OA}$$ = 2$$\vec{i}$$ – $$\vec{j}$$ + $$\vec{k}$$, $$\vec{OB}$$ = $$\vec{i}$$ – 3$$\vec{j}$$ – 5$$\vec{k}$$ then |$$\vec{OA}$$ × $$\vec{OB}$$| =
(a) 8$$\vec{i}$$ + 11$$\vec{j}$$ – 5$$\vec{k}$$
(b) $$\sqrt{210}$$
(c) sin θ
(d) $$\sqrt{40}$$

Answer: (b) $$\sqrt{210}$$

Question 30.
If |a| = |b| = |$$\vec{a}$$ + $$\vec{b}$$| = 1 then |$$\vec{a}$$ – $$\vec{b}$$| is equal to
(a) 1
(b) √3
(c) 0
(d) None of these

Question 31.
If $$\vec{a}$$ and $$\vec{b}$$ are any two vector then ($$\vec{a}$$ × $$\vec{b}$$)² is equal to
(a) ($$\vec{a}$$)²($$\vec{b}$$)² – ($$\vec{a}$$.$$\vec{b}$$)²
(b) ($$\vec{a}$$)² ($$\vec{b}$$)² + ($$\vec{a}$$.$$\vec{b}$$)²
(c) ($$\vec{a}$$.$$\vec{b}$$)²
(d) ($$\vec{a}$$)²($$\vec{b}$$)²

Answer: (a) ($$\vec{a}$$)²($$\vec{b}$$)² – ($$\vec{a}$$.$$\vec{b}$$)²

Question 32.
If $$\hat{a}$$ and $$\hat{b}$$ be two unit vectors and 0 is the angle between them, then |$$\hat{a}$$ – $$\hat{b}$$| is equal to
(a) sin $$\frac{θ}{2}$$
(b) 2 sin $$\frac{θ}{2}$$
(c) cos $$\frac{θ}{2}$$
(d) 2 cos $$\frac{θ}{2}$$

Answer: (b) 2 sin $$\frac{θ}{2}$$

Question 33.
The angle between the vector 2$$\hat{i}$$ + 3$$\hat{j}$$ + $$\hat{k}$$ and 2$$\hat{i}$$ – $$\hat{j}$$ – $$\hat{k}$$ is
(a) $$\frac{π}{2}$$
(b) $$\frac{π}{4}$$
(c) $$\frac{π}{3}$$
(d) 0

Answer: (a) $$\frac{π}{2}$$

Question 34.
If $$\vec{a}$$ = $$\hat{i}$$ – $$\hat{j}$$ + $$\hat{k}$$, $$\vec{b}$$ = $$\hat{i}$$ + 2$$\hat{j}$$ – $$\hat{k}$$, $$\vec{c}$$ = 3$$\hat{i}$$ – p$$\hat{j}$$ – 5$$\hat{k}$$ are coplanar then P =
(a) 6
(b) -6
(c) 2
(d) -2

Question 35.
The distance of the point (- 3, 4, 5) from the origin
(a) 50
(b) 5√2
(c) 6
(d) None of these

Question 36.
If $$\vec{AB}$$ = 2$$\hat{i}$$ + $$\hat{j}$$ – 3$$\hat{k}$$ and the co-ordinates of A are (1, 2, -1) then coordinate of B are
(a) (2, 2, -3)
(b) (3, 2, -4)
(c) (4, 2, -1)
(d) (3, 3, -4)

Question 37.
If $$\vec{b}$$ is a unit vector in xy-plane making an angle of $$\frac{π}{4}$$ with x-axis. then $$\vec{b}$$ is equal to
(a) $$\hat{i}$$ + $$\hat{j}$$
(b) $$\vec{i}$$ – $$\vec{j}$$
(c) $$\frac{\vec{i}+\vec{j}}{√2}$$
(d) $$\frac{\vec{i}-\vec{j}}{√2}$$

Answer: (c) $$\frac{\vec{i}+\vec{j}}{√2}$$

Question 38.
$$\vec{a}$$ = 2$$\hat{i}$$ + $$\hat{j}$$ – 8$$\hat{k}$$ and $$\vec{b}$$ = $$\hat{i}$$ + 3$$\hat{j}$$ – 4$$\hat{k}$$ then the magnitude of $$\vec{a}$$ + $$\vec{b}$$ is equal to
(a) 13
(b) $$\frac{13}{4}$$
(c) $$\frac{3}{13}$$
(d) $$\frac{4}{13}$$

Question 39.
The vector in 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 40.
The position vector of the point which divides the join of points 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) $$\frac{7\vec{a}-8\vec{b}}{2}$$
(c) $$\frac{3\vec{a}}{2}$$
(d) $$\frac{5\vec{a}}{4}$$

Answer: (d) $$\frac{5\vec{a}}{4}$$

Question 41.
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 42.
Find the value of λ such that the vectors $$\vec{a}$$ = 2$$\hat{i}$$ + λ$$\hat{j}$$ + $$\hat{k}$$ 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 43.
The value of λ for which the vectors 3$$\hat{i}$$ – 6$$\hat{j}$$ + $$\hat{k}$$ and 2$$\hat{i}$$ – 4$$\hat{j}$$ + λ$$\hat{k}$$ 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 44.
The vectors 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 45.
For any vector $$\vec{a}$$ the value of ($$\vec{a}$$ × $$\vec{i}$$)² + ($$\vec{a}$$ × $$\hat{j}$$)² + ($$\vec{a}$$ × $$\hat{k}$$)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 46.
If |$$\vec{a}$$| = 10, |$$\vec{b}$$| = 2 and $$\vec{a}$$.$$\vec{b}$$ = 12, then the value of |$$\vec{a}$$ × $$\vec{b}$$| is
(a) 5
(b) 10
(c) 14
(d) 16

Question 47.
The vectors λ$$\hat{i}$$ + $$\hat{j}$$ + 2$$\hat{k}$$, $$\hat{i}$$ + λ$$\hat{j}$$ – $$\hat{k}$$ and 2$$\hat{i}$$ – $$\hat{j}$$ + λ$$\hat{k}$$ are coplanar if
(a) λ = -2
(b) λ = 0
(c) λ = 1
(d) λ = -1

Question 48.
If $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ are unit vectors such that $$\vec{a}$$ + $$\vec{b}$$ + $$\vec{c}$$ = $$\vec{0}$$, then the value of $$\vec{a}$$.$$\vec{b}$$ + $$\vec{b}$$.$$\vec{c}$$ + $$\vec{c}$$.$$\vec{a}$$
(a) 1
(b) 3
(c) –$$\frac{3}{2}$$
(d) None of these

Answer: (c) –$$\frac{3}{2}$$

Question 49.
Projection vector of $$\vec{a}$$ on $$\vec{b}$$ is
(a) ($$\frac{\vec{a}.\vec{b}}{|\vec{b}|^2}$$)$$\vec{b}$$
(b) $$\frac{\vec{a}.\vec{b}}{|\vec{b}|}$$
(c) $$\frac{\vec{a}.\vec{b}}{|\vec{a}|}$$
(d) ($$\frac{\vec{a}.\vec{b}}{|\vec{a}|^2}$$)$$\hat{b}$$

Answer: (b) $$\frac{\vec{a}.\vec{b}}{|\vec{b}|}$$

Question 50.
If $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ are three vectors such that $$\vec{a}$$ + $$\vec{b}$$ + $$\vec{c}$$ = 5 and |$$\vec{a}$$| = 2, |$$\vec{b}$$| = 3, |$$\vec{c}$$| = 5, then the value of $$\vec{a}$$.$$\vec{b}$$ +$$\vec{b}$$.$$\vec{c}$$ + $$\vec{c}$$.$$\vec{a}$$ is
(a) 0
(b) 1
(c) -19
(d) 38

Question 51.
If |$$\vec{a}$$| 4 and – 3 ≤ λ ≤ 2, then the range of |λ$$\vec{a}$$| is
(a) [0, 8]
(b) [-12, 8]
(c) [0, 12]
(d) [8, 12]

Question 52.
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

Question 53.
If ($$\frac{1}{2}$$, $$\frac{1}{3}$$, n) are the direction cosines of a line, then the value of n is
(a) $$\frac{\sqrt{23}}{6}$$
(b) $$\frac{23}{6}$$
(c) $$\frac{2}{3}$$
(d) –$$\frac{3}{2}$$

Answer: (a) $$\frac{\sqrt{23}}{6}$$

Question 54.
Find the magnitude of vector 3$$\hat{i}$$ + 2$$\hat{j}$$ + 12$$\hat{k}$$
(a) $$\sqrt{157}$$
(b) 4$$\sqrt{11}$$
(c) $$\sqrt{213}$$
(d) 9√3

Answer: (a) $$\sqrt{157}$$

Question 55.
Three points (2, -1, 3), (3, – 5, 1) and (-1, 11, 9) are
(a) Non-collinear
(b) Non-coplanar
(c) Collinear
(d) None of these

Question 56.
The vectors 3$$\hat{i}$$ + 5$$\hat{j}$$ + 2$$\hat{k}$$, 2$$\hat{i}$$ – 3$$\hat{j}$$ – 5$$\hat{k}$$ and 5$$\hat{i}$$ + 2$$\hat{j}$$ – 3$$\hat{k}$$ form the sides of
(a) Isosceles triangle
(b) Right triangle
(c) Scalene triangle
(d) Equilateral triangle

Question 57.
The points with position vectors 60$$\hat{i}$$ + 3$$\hat{j}$$, 40$$\hat{i}$$ – 8$$\hat{j}$$ and a$$\hat{i}$$ – 52$$\hat{j}$$ are collinear if
(a) a = -40
(b) a = 40
(c) a = 20
(d) None of these

Question 58.
The ratio in which 2x + 3y + 5z = 1 divides the line joining the points (1, 0, -3) and (1, -5, 7) is
(a) 5 : 3
(b) 3 : 2
(c) 2 : 1
(d) 1 : 3

Question 59.
If O is origin and C is the mid point of A (2, -1) and B (-4, 3) then the value of $$\bar{OC}$$ is
(a) $$\hat{i}$$ + $$\hat{j}$$
(b) $$\hat{i}$$ – $$\hat{j}$$
(c) –$$\hat{i}$$ + $$\hat{j}$$
(d) –$$\hat{i}$$ – $$\hat{j}$$

Answer: (c) –$$\hat{i}$$ + $$\hat{j}$$

Question 60.
If ABCDEF is regular hexagon, then $$\vec{AD}$$ + $$\vec{EB}$$ + $$\vec{FC}$$ is equal
(a) 0
(b) 2$$\vec{AB}$$
(c) 3$$\vec{AB}$$
(d) 4$$\vec{AB}$$

Answer: (d) 4$$\vec{AB}$$

Question 61.
If $$\vec{a}$$ = $$\hat{i}$$ + $$\hat{j}$$ + $$\hat{k}$$, $$\vec{b}$$ = 2$$\hat{i}$$ – 4$$\hat{k}$$, $$\vec{c}$$ = $$\hat{i}$$ + λ$$\hat{j}$$ + 3$$\hat{j}$$ are coplanar, then the value of λ is
(a) $$\frac{5}{2}$$
(b) $$\frac{3}{5}$$
(c) $$\frac{7}{3}$$
(d) –$$\frac{5}{3}$$

Answer: (d) –$$\frac{5}{3}$$

Question 62.
The vectors $$\vec{a}$$ = x$$\hat{i}$$ – 2$$\hat{j}$$ + 5$$\hat{k}$$ and $$\vec{b}$$ = $$\hat{i}$$ + y$$\hat{j}$$ – z$$\hat{k}$$ are collinear, if
(a) x = 1, y = -2, z = -5
(b) x = $$\frac{3}{2}$$, y = -4, z = -10
(c) x = $$\frac{3}{2}$$, y = 4, z = 10
(d) All of these

Question 63.
The vectors (x, x + 1, x + 2), (x + 3, x + 4, x + 5) and (x + 6, x + 7, x + 8) are coplanar for
(a) all values of x
(b) x < 0
(c) x ≤ 0
(d) None of these

Answer: (a) all values of x

Question 64.
The vectors $$\vec{AB}$$ = 3$$\hat{i}$$ +4$$\hat{k}$$ and $$\vec{AC}$$ = 5$$\hat{i}$$ – 2$$\hat{j}$$ + 4$$\hat{k}$$ are the sides of ΔABC. The length of the median through A is
(a) $$\sqrt{18}$$
(b) $$\sqrt{72}$$
(c) $$\sqrt{33}$$
(d) $$\sqrt{288}$$

Answer: (c) $$\sqrt{33}$$

Question 65.
The summation of two unit vectors is a third unit vector, then the modulus of the difference of the unit vector is
(a) √3
(b) 1 – √3
(c) 1 + √3
(d) -√3