Vector Algebra

CBSE Class 12 Maths · 11 questions · 26 marks

Vectors capture quantities that have both magnitude and direction, and this chapter builds the algebra needed to manipulate them — addition, scalar multiplication, and two very different products. The dot product measures how aligned two vectors are and gives angles and projections, while the cross product produces a new vector perpendicular to both, whose magnitude equals the area it sweeps out. Together they set up the geometry used throughout three-dimensional coordinate geometry in the next chapter.

Quick revision

A vector a = a1î + a2ĵ + a3k̂ has magnitude |a| = √(a1²+a2²+a3²); the unit vector in its direction is â = a/|a|.
The position vector of the point dividing the segment joining A(a) and B(b) internally in the ratio m:n is (m·b + n·a)/(m+n).
Dot product: a·b = |a||b|cosθ = a1b1+a2b2+a3b3; a and b are perpendicular exactly when a·b = 0.
Cross product: a×b = |a||b|sinθ n̂, where n̂ is the unit vector perpendicular to both a and b; |a×b| equals the area of the parallelogram with sides a and b.
In component form, a×b is computed as the determinant with rows î ĵ k̂, (a1,a2,a3), (b1,b2,b3).
Scalar projection of a on b = (a·b)/|b|; this is the length of the shadow a casts along the direction of b.
Two nonzero vectors a and b are parallel (collinear) exactly when a = λb for some scalar λ, i.e. their components are proportional.
The scalar triple product [a b c] = a·(b×c) gives the volume of the parallelepiped formed by a, b, c; three vectors are coplanar exactly when [a b c] = 0.
Direction cosines l, m, n of a vector satisfy l² + m² + n² = 1, and are the cosines of the angles the vector makes with the x, y and z axes.

Important questions

1
MCQ1 mark
The magnitude of the vector a = 3î - 4ĵ + 12k̂ is:
1. A.13
2. B.19
3. C.5
4. D.7
Answer
The magnitude is 13. |a| = √(3² + (-4)² + 12²) = √(9+16+144) = √169 = 13.
2
MCQ1 mark
If a·b = 0 for two nonzero vectors a and b, then a and b are:
1. A.Perpendicular
2. B.Parallel
3. C.Equal
4. D.Collinear
Answer
a and b are perpendicular. Since a·b = |a||b|cosθ and both |a| and |b| are nonzero, a·b = 0 forces cosθ = 0, so θ = 90°, meaning a and b are perpendicular.
3
MCQ1 mark
The area of the parallelogram with adjacent sides a = î + ĵ and b = î - ĵ is:
1. A.2 sq units
2. B.1 sq unit
3. C.4 sq units
4. D.√2 sq units
Answer
The area is 2 sq units. a×b = determinant with rows î ĵ k̂, (1,1,0), (1,-1,0) = î(1×0-0×(-1)) - ĵ(1×0-0×1) + k̂(1×(-1)-1×1) = 0î - 0ĵ - 2k̂. |a×b| = √(0²+0²+(-2)²) = 2, which is the area of the parallelogram.
4
MCQ1 mark
The projection of a = 2î + 3ĵ + k̂ on b = î + 2ĵ + 2k̂ is:
1. A.10/3
2. B.10
3. C.3/10
4. D.30
Answer
The projection is 10/3. a·b = 2(1) + 3(2) + 1(2) = 2+6+2 = 10. |b| = √(1²+2²+2²) = √9 = 3. Projection of a on b = (a·b)/|b| = 10/3.
5
Assertion & Reason1 mark
Assertion (A): The vectors a = 2î + 3ĵ - k̂ and b = -6î - 9ĵ + 3k̂ are parallel. Reason (R): Two vectors are parallel if and only if one is a scalar multiple of the other.
1. A.Both A and R are true and R is the correct explanation of A
2. B.Both A and R are true but R is not the correct explanation of A
3. C.A is true but R is false
4. D.A is false but R is true
Answer
Both A and R are true and R is the correct explanation of A. Reason R correctly states the general condition for parallel vectors. For Assertion A, notice b = -3a, since -3(2î+3ĵ-k̂) = -6î-9ĵ+3k̂, exactly matching b. Since b is a scalar multiple of a, they are parallel — confirming A precisely via the criterion in R.
6
Short answer2 marks
Find the unit vector in the direction of a = 4î - 3ĵ + 12k̂.
7
Short answer2 marks
Find λ if the vectors a = 2î + λĵ + k̂ and b = 4î - 2ĵ - 2k̂ are perpendicular.
8
Short answer3 marks
Find the angle between the vectors a = î + ĵ - k̂ and b = î - ĵ + k̂.
9
Long answer5 marks
Find a×b if a = î + 2ĵ + 3k̂ and b = 3î - ĵ + 2k̂, and hence find a unit vector perpendicular to both a and b.
10
Long answer5 marks
Show that the vectors a = î - 2ĵ + 3k̂, b = -2î + 3ĵ - 4k̂ and c = î - 3ĵ + 5k̂ are coplanar.
11
Case study4 marks
Read the following and answer the questions that follow: A drone's displacement from its launch point is represented by the vector d = 3î + 4ĵ + 12k̂ (in metres), and a gust of wind exerts a force vector f = î - 2ĵ + 2k̂ (in newtons) on it. (a) Find |d|, the magnitude of the displacement. (b) Find the unit vector in the direction of d. (c) Find d·f. (d) Find the scalar projection of f on d.

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Vector Algebra

CBSE Class 12 Maths · 11 questions · 26 marks

Quick revision

A vector a = a1î + a2ĵ + a3k̂ has magnitude |a| = √(a1²+a2²+a3²); the unit vector in its direction is â = a/|a|.

The position vector of the point dividing the segment joining A(a) and B(b) internally in the ratio m:n is (m·b + n·a)/(m+n).

Dot product: a·b = |a||b|cosθ = a1b1+a2b2+a3b3; a and b are perpendicular exactly when a·b = 0.

Cross product: a×b = |a||b|sinθ n̂, where n̂ is the unit vector perpendicular to both a and b; |a×b| equals the area of the parallelogram with sides a and b.

In component form, a×b is computed as the determinant with rows î ĵ k̂, (a1,a2,a3), (b1,b2,b3).

Scalar projection of a on b = (a·b)/|b|; this is the length of the shadow a casts along the direction of b.

Two nonzero vectors a and b are parallel (collinear) exactly when a = λb for some scalar λ, i.e. their components are proportional.

The scalar triple product [a b c] = a·(b×c) gives the volume of the parallelepiped formed by a, b, c; three vectors are coplanar exactly when [a b c] = 0.

Direction cosines l, m, n of a vector satisfy l² + m² + n² = 1, and are the cosines of the angles the vector makes with the x, y and z axes.

Important questions

MCQ1 mark

The magnitude of the vector a = 3î - 4ĵ + 12k̂ is:

A.13
B.19
C.5
D.7

MCQ1 mark

If a·b = 0 for two nonzero vectors a and b, then a and b are:

A.Perpendicular
B.Parallel
C.Equal
D.Collinear

MCQ1 mark

The area of the parallelogram with adjacent sides a = î + ĵ and b = î - ĵ is:

A.2 sq units
B.1 sq unit
C.4 sq units
D.√2 sq units

MCQ1 mark

The projection of a = 2î + 3ĵ + k̂ on b = î + 2ĵ + 2k̂ is:

A.10/3
B.10
C.3/10
D.30

Assertion & Reason1 mark

Assertion (A): The vectors a = 2î + 3ĵ - k̂ and b = -6î - 9ĵ + 3k̂ are parallel. Reason (R): Two vectors are parallel if and only if one is a scalar multiple of the other.

A.Both A and R are true and R is the correct explanation of A
B.Both A and R are true but R is not the correct explanation of A
C.A is true but R is false
D.A is false but R is true

Short answer2 marks

Find the unit vector in the direction of a = 4î - 3ĵ + 12k̂.

Short answer2 marks

Find λ if the vectors a = 2î + λĵ + k̂ and b = 4î - 2ĵ - 2k̂ are perpendicular.

Short answer3 marks

Find the angle between the vectors a = î + ĵ - k̂ and b = î - ĵ + k̂.

Long answer5 marks

Find a×b if a = î + 2ĵ + 3k̂ and b = 3î - ĵ + 2k̂, and hence find a unit vector perpendicular to both a and b.

Long answer5 marks

Show that the vectors a = î - 2ĵ + 3k̂, b = -2î + 3ĵ - 4k̂ and c = î - 3ĵ + 5k̂ are coplanar.

Case study4 marks

Read the following and answer the questions that follow: A drone's displacement from its launch point is represented by the vector d = 3î + 4ĵ + 12k̂ (in metres), and a gust of wind exerts a force vector f = î - 2ĵ + 2k̂ (in newtons) on it. (a) Find |d|, the magnitude of the displacement. (b) Find the unit vector in the direction of d. (c) Find d·f. (d) Find the scalar projection of f on d.