Please refer to Vector Algebra Class 12 Mathematics Exam Questions provided below. These questions and answers for Class 12 Mathematics have been designed based on the past trend of questions and important topics in your Class 12 Mathematics books. You should go through all Class 12 Mathematics Important Questions provided by our teachers which will help you to get more marks in upcoming exams.

**Class 12 Mathematics** **Exam Questions Vector Algebra**

Class 12 Mathematics students should read and understand the important questions and answers provided below for Vector Algebra which will help them to understand all important and difficult topics.

**Very Short Answer Type Questions**

**Question.****Find the direction cosine of the vector a̅, = î + 2ĵ + k̂+ 2 +3 . î ĵ k̂ a̅, b̅,Answer.**

**Question.****If a vector has direction ratios 2, -1, -2 then what are its direction cosines?Answer.** Here DR’s are a = 2 , b = -1 , c = -2

**Question.****Write the value of p for which the vectors a̅, = î + 2ĵ + 9k̂ and b̅, = î – 2pĵ + 3k̂are parallel ?Answer.** DR’s of I

^{st }vector are a

_{1}= 3, b

_{1}= 2, c

_{1}= 9

DR’s of II

^{nd }vector are a

_{2}= 1, b

_{2}= 2p, c

_{2}= 3

**Question.****Write the value of so that the vector b̅, = 2î + λĵ + 3k̂ and b̅, = î – ĵ + 3k̂ are perpendicular to each other.Answer.** If vectors a̅, and b̅ are perpendiculars then

a

_{1}-a

_{2}+b

_{1}+b

_{2}+c

_{1}+c

_{2}= 0

2 x 1 λ x (-2) + 3 x 3 = 0

**Question.****find a unit vector in the direction of a̅, = î – 2ĵ + 3k̂Answer.**

**Question.****Find a vector in the direction of a̅, = î – 2ĵ + 3k̂ that has magnitude 7 units.Answer.**

**Question.****Find a vector of magnitude 5 units and parallel to the resultant of b̅, = î – ĵ + k̂Answer.** r̅, = a̅ + b̅

= 3î + ĵ

Unit Vector in the direction of

Vector of magnitude 5 units and parallel to the resultant of a̅, and b̅ is

**Question.****Find the position vector of a point which divides the join of points with position vector a̅, – 2b̅ and 2a̅, + b̅ externally in the ratio 2: 1.Answer.**

**Question.****Find the position vector of a point R which divides the line joining the two points P and Q whose position vectors are î + 2ĵ – k̂ and of – î + ĵ + k̂ respectively in the ratio 2: 1(i) Internally (ii) externallyAnswer.** Vector that divides internally is

Vector that divides Externally is

**Question.****Write the projection of vector a̅, = 7î + ĵ – 4k̂ on the vector b̅, = 2î – 6ĵ + 3k̂Answer.** the projection of vector a̅ on the vector b̅

**1. If a unit vector A̅ makes angles π/3 with î, π/4 with ĵ, and an acute angle θ with k̂, then find the value of θ.**

**Answer.**

**2. Find the sum of the following vectors.**

**Answer.**

**3. Find a unit vector in the direction of the sum of the vectors**

**Answer.**

**4 L and M are two points with position vectors 2a̅, − b̅, and a̅, + 2b̅, respectively. What is the position vector of a point N which divides the line segment LM in the ratio 2 : 1 externally ?****Answer.** Required position vector

**5. Find the value of |a̅×b̅ ^{2}| + | a̅, b̅^{2}|| a̅ | = 5 and | b̅ | = 4.**

**Answer.**

**6. Find the area of a parallelogram whose** **adjacent sides are represented by the vectors** **î, − 3 k̂ and 2ĵ, + k̂.****Answer.**

**7. Find the projection of the vector**

**Answer.**

**8. If a̅, and b̅ are unit vectors, then find the angle between a̅ and b̅ , given that ( √2**

**a̅ – b̅ ) is a unit vector**

**Answer.**Let q be the angle between the unit vectors a̅, and b̅

**9. Find the angle between x-axis and the** **vector î, + ĵ, + k̂** **Answer.** Here, a̅ = î, + ĵ, + k̂ and vector along x-axis is ĵ.

∴ Angle between a̅ and î, is given by

**10. If a̅, and b̅ are two vectors such that** **| a̅ + b̅| = | a̅ |** **then prove that vector 2a̅+ b̅ is perpendicular to** **vector** **Answer.**

**Short Answer Type Questions -I**

**11. X and Y are two points with position vectors 3a̅ + b̅ and a̅ − 3b̅ respectively. Write the position vector of a point Z which divides the line segment XY in the ratio 2 : 1 externally.** **Answer.** Position vector which divides the line segment joining points with position vectors 3a̅ + b̅ and a̅ − 3b̅ in the ratio 2 : 1 externally is given by

**12. Find a unit vector perpendicular to each of** **the vectors a̅ and b̅ and where a̅ = 5î + 6ĵ – 2k̂ and b̅ = 7î + 6ĵ – 2k̂****Answer.**

**13. show that for any two non zero vectors a̅ and b̅, |a̅ + b̅| = |a̅ – b̅| iff a̅ and b̅ are perpendicular vectors.Answer.** For any two non-zero vectors a̅ and b̅ , we have

**CASE STUDY :A cricket match is organized between two Clubs A and B for which a team from each club is chosen. Remaining players of Club A and Club B are respectively sitting on the plane represented by the equation **

**to cheer the team of their own clubs.Based on the above answer the following:**

**Question. The Cartesian equation of the plane on which players of Club A are seated is**a. 2𝑥−𝑦+𝑧=3

b. 2𝑥−𝑦+2𝑧=3

c. 2𝑥−𝑦+𝑧=−3

d. 𝑥−𝑦+𝑧=3

## Answer

A

** Question. The magnitude of the normal to the plane on which players of club B are seated, is**a. √15

b. √14

c. √17

d. √20

## Answer

B

**Question**. The intercept form of the equation of the plane on which players of Club B are seated is

## Answer

C

** Question. Which of the following is a player of Club B?**a. Player sitting at (1, 2, 1)

b. Player sitting at (0, 1, 2)

c. Player sitting at (1, 4, 1)

d. Player sitting at (1, 1, 2)

## Answer

D

**Question. The distance of the plane, on which players of Club B are seated, from the origin is**a. 8/√14 𝑢𝑛𝑖𝑡𝑠

b. 6/√14 𝑢𝑛𝑖𝑡𝑠

c. 7/√14 𝑢𝑛𝑖𝑡𝑠

d. 9/√14 𝑢𝑛𝑖𝑡𝑠

## Answer

A

**CASE STUDY :**

**The equation of motion of a missile are x = 3t, y = -4t, z = t, where the time ‘t’ is given in seconds, and the distance is measured in kilometres.**

Based on the above answer the following:

Based on the above answer the following:

**Question. What is the path of the missile?**a. Straight line

b. Parabola

c. Circle

d. Ellipse

## Answer

A

** Question. Which of the following points lie on the path of the missile?**a. (6, 8, 2)

b. (6, -8, -2)

c. (6, -8, 2)

d. (-6, -8, 2)

## Answer

C

** Question. At what distance will the rocket be from the starting point (0, 0, 0) in 5 seconds?**a. √550 kms

b. √650 kms

c. √450 kms

d. √750 kms

## Answer

B

** Question. If the position of rocket at a certain instant of time is (5, -8, 10), then what will be the height of the rocket from the ground? (The ground is considered as the xy – plane).**a. 12 km

b. 11 km

c. 20 km

d. 10 km

## Answer

D

** Question. At a certain instant of time, if the missile is above the sea level, where the equation of the surface of sea is given by 2x + y + 3z = 1 and the position of the missile at that instant of time is (1, 1, 2), then the image of the position of the rocket in the sea is**a. (−9/7,−1/7,−10/7)

b. (9/7,−1/7,−10/7)

c. (−9/7,1/7,−10/7)

d. (−9/7,−1/7,10/7)

## Answer

A