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Vector Algebra – Class 12 Mathematics notes, important questions and MCQs

NCERT Class 12 Mathematics, chapter 10. Notes, key terms, important questions with answers and MCQs.

Chapter overview

Vectors have magnitude and direction. Addition, scalar (dot) product and vector (cross) product solve problems on angles, projections and areas.

Basics

Position vector of (x, y, z): xî + yĵ + zk̂; magnitude √(x² + y² + z²).
Unit vector â = a/|a|; direction cosines l, m, n with l² + m² + n² = 1.
Section formula: (mb + na)/(m + n).

Products

Dot product a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃; a ⟂ b ⇔ a·b = 0.
Projection of a on b = (a·b)/|b|.
Cross product |a × b| = |a||b| sin θ; area of parallelogram = |a × b|; area of triangle = ½|a × b|; a ∥ b ⇔ a × b = 0.

Key terms

Unit vector: vector of magnitude 1
Collinear vectors: vectors parallel to the same line
Scalar triple product: [a b c] = a·(b × c), the volume of the parallelepiped formed by a, b, c; zero for coplanar vectors

Formulas

a·b = |a||b| cos θ
|a × b| = |a||b| sin θ
Projection = (a·b)/|b|

Important questions with answers

Find a unit vector in the direction of a = 2î + 3ĵ + k̂.2 marks

|a| = √14, â = (2î + 3ĵ + k̂)/√14.

Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5), C(1, 5, 5).3 marks

AB = î + 2ĵ + 3k̂, AC = 4ĵ + 3k̂. AB × AC = (6 − 12)î − (3 − 0)ĵ + (4 − 0)k̂ = −6î − 3ĵ + 4k̂. |AB × AC| = √61. Area = ½√61 square units.

If a + b + c = 0 with |a| = 3, |b| = 5, |c| = 7, find the angle between a and b. Also find the projection of b on a when a = î + 2ĵ + 2k̂ and b = 3î + ĵ + 2k̂ (separate vectors).5 marks

a + b = −c, so |a + b|² = |c|²: 9 + 25 + 2a·b = 49, a·b = 7.5. cos θ = 7.5/15 = 1/2, θ = 60°. Projection of b on a = (a·b)/|a| = (3 + 2 + 4)/3 = 3.

Quick revision

Dot product is a scalar; cross product is a vector.
Perpendicular ⇒ dot = 0.
Parallel ⇒ cross = 0.

Exam tips

î·î = 1, î·ĵ = 0; î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ.
Angle between vectors: cos θ = (a·b)/(|a||b|); always find magnitudes first.

Practice MCQs

Magnitude of 3î + 4ĵ is
1. 7
2. 25
3. 5
4. 1
Show answer
Answer: (c) 5. √(9 + 16).
î × ĵ equals
1. −k̂
2. k̂
3. 0
4. 1
Show answer
Answer: (b) k̂. Right-hand rule.
(î + 2ĵ + 3k̂)·(2î − ĵ + k̂) equals
1. 3
2. 5
3. 7
4. 1
Show answer
Answer: (a) 3. 2 − 2 + 3.
Vectors a and b are perpendicular if
1. a × b = 0
2. |a| = |b|
3. a·b = 0
4. a = b
Show answer
Answer: (c) a·b = 0. cos 90° = 0.
If |a| = 2, |b| = 3 and a·b = 3, the angle between a and b is
1. 30°
2. 45°
3. 60°
4. 90°
Show answer
Answer: (c) 60°. cos θ = 1/2.
Area of the parallelogram with adjacent sides î + ĵ and î − ĵ is
1. 1
2. √2
3. 2
4. 4
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Answer: (c) 2. |(î + ĵ) × (î − ĵ)| = |−2k̂| = 2.
Projection of 2î + 3ĵ + 2k̂ on î + 2ĵ + k̂ is
1. 5
2. 10
3. 6/√10
4. 10/√6
Show answer
Answer: (d) 10/√6. (2 + 6 + 2)/√6.
The value of λ for which 2î + λĵ + k̂ and î − 2ĵ + 3k̂ are perpendicular is
1. 1.5
2. 2.5
3. 2
4. −2.5
Show answer
Answer: (b) 2.5. 2 − 2λ + 3 = 0.

All chapters of Class 12 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.