Vector Algebra Class 12 Mathematics Important Questions

Please refer to Vector Algebra Class 12 Mathematics Important Questions with solutions provided below. These questions and answers have been provided for Class 12 Mathematics based on the latest syllabus and examination guidelines issued by CBSE, NCERT, and KVS. Students should learn these problem solutions as it will help them to gain more marks in examinations. We have provided Important Questions for Class 12 Mathematics for all chapters in your book. These Board exam questions have been designed by expert teachers of Standard 12.

Class 12 Mathematics Important Questions Vector Algebra

Very Short Answer Type Questions

Question. 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.

Question. Find the magnitude of the vector

Answer.

Question. Write a unit vector perpendicular to both the vectors a̅ = iˆ + jˆ + kˆ and b̅ = iˆ + jˆ.
Answer.

Question.

Answer.

Question. Write the projection of the vector iˆ + jˆ + kˆ along the vector ĵ
Answer. The projection of the vector (iˆ + jˆ + kˆ) along the vector

Question. Write the value of

Answer. We have

Question. Write the value of cosine of the angle which the vector a̅ = iˆ + jˆ + kˆ makes with y-axis.
Answer. Let θ be the angle between the vector

Question.

Answer. Let angle between the vectors a̅ and b̅ be θ.

Question. Find the area of a parallelogram whose adjacent sides are represented by the vectors 2iˆ − 3kˆ and 4 jˆ + 2kˆ
Answer.

Question. If a̅ and b̅ are unit vectors, then what is the angle between a̅ and b̅ so that √2 a̅ − b̅ is a unit vector ?
Answer. Let θ be the angle between the unit vectors a̅ and b̅

Question. If a̅ and b̅ are two unit vectors such that a̅ + b̅ is also a unit vector, then find the angle between a̅ and b̅.
Answer.

Question.

Answer.

Question. If a̅ and b̅ are perpendicular vectors,

Answer.

Question. Find the magnitude of the vector
Answer. The given vector, a̅ = 2î − 6 ĵ − 3k̂

Question.

Answer. The given vectors are

Question. Find the sum of the following vectors :

Answer. Required sum = a̅ + b̅ + c̅

Question. Find the sum of the following vectors :

Answer.

Question. If A, B and C are the vertices of a triangle ABC, then what is the value of

Answer. Let ABC be the given triangle.

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

Question. Write the position vector of the point which divides the join of points with position vectors

Answer. Required position vector

Short Answer Type Questions

Question. The two vectors ĵ + k̂ and 3î − ĵ + 4k̂ represent the two sides AB and AC, respectively of a ΔABC. Find the length of the median through A.
Answer. Take A to be as origin (0, 0, 0).
∴ Coordinates of B are (0, 1, 1) and coordinates of C are (3, –1, 4).

Question. Find a vector of magnitude 5 units and parallel to the resultant of the vectors a̅ = 2iˆ + 3jˆ − kˆ and b̅ = iˆ − 2 jˆ + kˆ.
Answer.

Question.

Answer.

Question.

Answer.

Question.

Answer.

Question. The scalar product of the vector a̅ = iˆ + jˆ + kˆ with a unit vector along the sum of vectors b̅ = 2iˆ + 4 jˆ − 5kˆ and c̅ = λiˆ + 2 jˆ + 3kˆ is equal to one. Find the value of λ and hence find the unit vector along b̅ + c̅
Answer.

Question. Find a unit vector perpendicular to both

Answer.

Question. If a̅ = 2iˆ − 3jˆ + kˆ ,b̅ = −iˆ + kˆ , c̅ = 2 jˆ − kˆ are three vectors, find the area of the parallelogram having diagonals (a̅ + b̅) and (b̅ + c̅).
Answer.

Question.

Answer.

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

Question.

Answer.

Question. Using vectors, find the area of the triangle ABC with vertices A(1, 2, 3), B(2, –1, 4) and C(4, 5, –1).
Answer. Given, ΔABC with vertices

Question.

Answer. We have,

Question. If the sum of two unit vectors a̅ and b̅ is a unit vector, show that the magnitude of their difference is √3
Answer.

Question. If two vectors a̅ and b̅ are such that

Answer.

Question.

Answer. Here,

Question. Show that the vectors a̅ ,b̅ ,c̅ are coplanar if and only if a̅+b̅ ,b̅+c̅ and c̅ + a̅ are coplanar.
Answer.

CASE STUDY:
The Indian coast guard, while patrolling, saw a suspicious boat with people. They were nowhere looking like fishermen. The coast guard were closely observing the movement of the boat for an opportunity to seize the boat. They observed that the boat is moving along a planar surface. At an instant of time, the coordinates of the position of the coast guard helicopter and the boat is (1, 3, 5) and (2, 5, 3) respectively.
Based on the above answer the following:

Question. If the line joining the positions of the helicopter and the boat is perpendicular to the plane in which the boat moves, then the equation of the plane is
a. -x + 2y – 2z = 6
b. x + 2y + 2z = 6
c. x + 2y – 2z = 6
d. x – 2y – 2z = 6

Question. If the coast guard decide to shoot the boat at that given instant of time, then what is the distance (in meters) that the bullet has to travel?
a. 5m
b. 3m
c. 6m
d. 4m

Question. If the coast guard decides to shoot the boat at that given instant of time, when the speed of bullet is 36m/sec, then what is the time taken for the bullet to travel and hit the boat?
a. 1/8𝑠𝑒𝑐𝑜𝑛𝑑𝑠
b. 1/14𝑠𝑒𝑐𝑜𝑛𝑑𝑠
c. 1/10𝑠𝑒𝑐𝑜𝑛𝑑𝑠
d. 1/12𝑠𝑒𝑐𝑜𝑛𝑑𝑠

Question. At that given instant of time, the equation of line passing through the positions of the helicopter and boat is

Question. At a different instant of time, the boat moves to a different position along the planar surface. What should be the coordinates of the location of the boat if the coast guard shoots the bullet along the line whose equation is

for the bullet to hit the boat?
a. (−8/3 ,19/3 ,−14/3)
b. (8/3 ,−19/3 ,−14/3)
c. (8/3 ,−19/3 ,14/3)
d. none of the above

CASE STUDY :
A mobile tower stands at the top of a hill. Consider the surface on which the tower stands as a plane having points A(1, 0, 2), B(3, -1, 1) and C(1, 2, 1) on it. The mobile tower is tied with 3 cables from the point A, B and C such that it stands vertically on the ground. The top of the tower is at the point (2, 3, 1) as shown in the figure.

Based on the above answer the following:

Question. The equation of the plane passing through the points A, B and C is
a. 3x – 2y + 4z = -11
b. 3x + 2y + 4z = 11
c. 3x – 2y – 4z = 11
d. -3x + 2y + 4z = -11

Question. The height of the tower from the ground is
a. 5/√29 𝑢𝑛𝑖𝑡𝑠
b. 7/√29 𝑢𝑛𝑖𝑡𝑠
c. 6/√29 𝑢𝑛𝑖𝑡𝑠
d. 8/√29 𝑢𝑛𝑖𝑡𝑠

Question. The equation of the perpendicular line drawn from the top of the tower to the ground is

Question. The coordinates of the foot of the perpendicular drawn from the top of the tower to the ground are
a. (43/29,−77/29,−9/29)
b. (9/7,−1/7,−10/7)
c. (−43/29,77/29,−9/29)
d. (43/29,77/29,9/29)

Question. The area of Δ 𝐴𝐵𝐶 is
a. √29/4 sq. units
b. √29/2 sq. units
c. √39/2 sq. units
d. √39/4 sq. units