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Mathematics — Class 12 — L34: Vectors

NIOS Code 311 · Module 9 · Vectors and 3D Geometry

Notes extracted from NIOS Mathematics Course (311), Lesson 34 — Vectors (ilovepdf_merged (6).pdf). Content covers sections 34.1–34.x.

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Overview — Vectors (L34)

A vector has magnitude and direction. Represented as AB⃗ or a⃗. Free vectors may be translated without changing them.

|a⃗| magnitude · â = a⃗/|a⃗| unit vector

Zero vector has magnitude 0

Directed segment as a vector

34.1 Operations

Addition by triangle/parallelogram law. Scalar multiple changes length (and reverses direction if scalar < 0). Position vector of a point A is OA⃗.

34.2 Dot and cross products

a·b = |a||b| cos θ · a×b is perpendicular to both with |a×b|=|a||b|sin θ

Dot scalar · cross vector

a·b = 0 ⇒ perpendicular (nonzero vectors).
a×b = 0 ⇒ parallel.

Right-hand rule: Direction of a×b follows the right-hand screw rule.

MCQ Quiz — L34 Vectors

0 / 10 correct

Flashcards — L34

1 / 16

Golden Rules — L34 Vectors

Most exam-important points from this chapter:

Know the definitions of L34

Start every answer with the key definition or standard form from Vectors.

Write the formula first

Copy the formula, then substitute values; never jump to the number alone.

Watch conditions

Check domain, quadrant, non-zero denominators, and applicability of the theorem.

Master result 1

Memorise and apply: After studying this lesson, you will be able to :

Master result 2

Memorise and apply: define and understand direction cosines and direction ratios of a vector.

|a| magnitude

unit â

a·b = |a||b|cosθ

a×b

|a×b|=|a||b|sinθ

i j k

1. Formulas & Definitions

Full Ch 34 — Vectors study guide: definitions, derivations, why each formula works, history, deep understanding, pencil diagrams, step-by-step use, and 4-level solved examples.

a · b = |a| |b| cos θ

Definition: Dot (scalar) product.

Derivation

From components a₁b₁+a₂b₂+a₃b₃ equals that form.

Variables

θ angle between · scalar result

Why it works

Measures alignment; zero iff perpendicular (nonzero vectors).

Historical context

Gibbs / Heaviside vector calculus.

Deep understanding

Work = F·d is classic application.

2. Diagrams & Visuals

Pencil sketch · labelled · step-by-step breakdown below

Find magnitudes and cosθ or use components
Multiply
Result scalar
Check ⊥ if 0

3. Solved Examples

Basic

Q: i·j

Solution: 0

Answer: 0

Intermediate

Q: (1,2,3)·(4,−5,6)

Solution: 4−10+18=12

Answer: 12

Advanced

Q: Angle: a·b=|a||b|cosθ

Solution: cosθ=(a·b)/(|a||b|)

Answer: Use arccos

Exam

Q: a⊥b iff?

Solution: a·b=0

Answer: a · b = 0

a × b (vector product)

Definition: Vector perpendicular to both; |a×b|=|a||b|sinθ; right-hand rule.

Derivation

Determinant of i j k with components.

Variables

Result vector · anti-commutative a×b=−b×a

Why it works

Area of parallelogram = |a×b|.

Historical context

Gibbs vector algebra.

Deep understanding

a×a=0; parallel vectors give zero cross.

2. Diagrams & Visuals

Pencil sketch · labelled · step-by-step breakdown below

Write determinant i j k
Compute components
Check direction RHS
Magnitude for area

3. Solved Examples

Basic

Q: i×j

Solution: k

Answer: k

Intermediate

Q: (1,0,0)×(0,1,0)

Solution: (0,0,1)

Answer: k

Advanced

Q: |(1,2,0)×(3,4,0)|

Solution: |0i−0j−2k|=2

Answer: 2

Exam

Q: a×b = 0 means?

Solution: Parallel (or zero)

Answer: Parallel vectors

Unit vector â = a / |a|

Definition: Vector of length 1 in direction of a (a≠0).

Derivation

Scale by reciprocal of magnitude.

Variables

|a| ≠ 0

Why it works

Direction without magnitude.

Historical context

Standard normalisation.

Deep understanding

DC of a line are components of a unit vector along it.

2. Diagrams & Visuals

Pencil sketch · labelled · step-by-step breakdown below

Compute |a|
Divide each component
Verify magnitude 1
Use as direction

3. Solved Examples

Basic

Q: Unit of (3,0,0)

Solution: (1,0,0)

Answer: (1, 0, 0)

Intermediate

Q: Unit of (2,−1,2)

Solution: (2/3,−1/3,2/3)

Answer: (2/3, −1/3, 2/3)

Advanced

Q: Vector length 5 along (1,2,2)

Solution: 5·(1,2,2)/3

Answer: (5/3, 10/3, 10/3)

Exam

Q: â · â = ?

Solution: 1

Answer: 1

5. Special Features & Extras

Complete study guide — exam tips, common mistakes, memory aids, and quick reference.

Exam Tips & Tricks

Dot → scalar; cross → vector.
i×j=k cyclic.
Check magnitudes before angle.

Common Student Mistakes

Treating cross product as commutative
Dot product as vector
Forgetting sin vs cos

Memory Aids & Mnemonics

Dot: cos (projection). Cross: sin (area).

i j k cyclic cross.

Which Formula When?

Angle/projection/work → dot
Perp vector/area → cross
Direction only → unit

Quick reference box

• a·b=|a||b|cosθ · |a×b|=|a||b|sinθ · â=a/|a|

PYQ — Previous Year Questions

English questions from NIOS Mathematics (311) public / sample papers (Sample 2024, Apr 2024, Oct 2024, Apr 2025; board-style where scanned papers had no extractable text). Mapped exclusively to this chapter — no cross-chapter duplicates. Use Model Answer for the marking key and Explanation for working.

L34 — Vectors

5 question(s) · Sources: Board-style dot, Board-style magnitude, Sample QP 2024

PYQ1. Which of the following is a vector quantity?

(A) Mass
(B) Force
(C) Time
(D) Length

1 mark(s) · MCQ · Sample QP 2024 · Q15(i)

Model Answer

1) Force

Explanation

1) Force has magnitude and direction; mass, time and length are scalars.

PYQ2. If vectors 3î + λĵ + k̂ and 2î − ĵ + 8k̂ are perpendicular, find λ.

1 mark(s) · SA · Sample QP 2024 · Q19

Model Answer

1) λ = 14

Explanation

1) Dot product 3·2 + λ(−1) + 1·8 = 0 ⇒ 6 − λ + 8 = 0 ⇒ λ = 14.

PYQ3. If a⃗ = 2î − ĵ + 2k̂ and b⃗ = −î + ĵ − k̂, find the unit vector in the direction of a⃗+b⃗.

2 mark(s) · SA · Sample QP 2024 · Q17(ii) OR concept

Model Answer

1) (1/√2) î + (1/√2) k̂ [since a+b = î + k̂, |a+b|=√2]

Explanation

1) a+b = î + k̂; |a+b|=√2; unit vector = (î+k̂)/√2.

PYQ4. Find |2î − ĵ + 2k̂|.

1 mark(s) · SA · Board-style magnitude

Model Answer

1) 3

Explanation

1) √(4+1+4)=3.

PYQ5. Find (î+2ĵ)·(3î−ĵ).

1 mark(s) · SA · Board-style dot

Model Answer

1) 1

Explanation

1) 1·3 + 2·(−1) = 3−2=1.

Problem Solving — L34 Vectors

10 English exam-style problems for this chapter (NIOS Mathematics 311). Each question states the formula, gives full step-by-step working with the final answer, plus formal/easy explanations. English only.

Question 1 of 10Magnitude

Find |a⃗| if a⃗=2î−ĵ+2k̂.

|a|=√(a₁²+a₂²+a₃²)

1) √(4+1+4)=√9=3.

Answer: 3

Formula used

|a|=√(a₁²+a₂²+a₃²)

Textbook formal language

Euclidean norm of components.

Easy language (same calculation)

2,1,2 is 3-4-5 style with 3.

Why this formula

Unit vector â=a/|a|=(2/3)i−(1/3)j+(2/3)k.

Exam tip

Squares of components.

Common mistakes

√5
9

Question 2 of 10Dot

Find a·b for a=i+2j, b=3i−j.

a·b=|a||b|cosθ

1) 1·3+2·(−1)=3−2=1.

Answer: 1

Formula used

a·b=|a||b|cosθ

Textbook formal language

Sum of products of components.

Easy language (same calculation)

3 plus −2.

Why this formula

θ = cos⁻¹(1/(|a||b|)) if angle needed.

Exam tip

Include all components present.

Common mistakes

5
−1

Question 3 of 10Perp

For what k is a⃗ = î+2ĵ+k k̂ perpendicular to b⃗ = 2î−ĵ+k̂?

a·b=0

1) a·b = 1·2 + 2·(−1) + k·1 = 2 − 2 + k = k.

2) Set a·b = 0 ⇒ k = 0.

Answer: k = 0

Formula used

a·b=0

Textbook formal language

Dot product zero for perpendicular nonzero vectors.

Easy language (same calculation)

2−2+k=0 ⇒ k=0.

Why this formula

Check not both zero vectors.

Exam tip

k is the coefficient of k̂ in a.

Common mistakes

k=2
k=−2

Question 4 of 10Cross mag

If |a|=3, |b|=4, θ=90°, find |a×b|.

|a×b|=|a||b|sinθ

1) |a×b|=3·4·sin90°=12.

Answer: 12

Formula used

|a×b|=|a||b|sinθ

Textbook formal language

Perpendicular vectors: magnitude product.

Easy language (same calculation)

sin90=1 so 12.

Why this formula

Direction by right-hand rule (not asked).

Exam tip

If θ=0, cross product zero.

Common mistakes

Question 5 of 10Unit

Unit vector along 3i−6j+6k.

â=a/|a|

1) |a|=√(9+36+36)=√81=9.

2) â=(1/3)i−(2/3)j+(2/3)k.

Answer: (1/3)i − (2/3)j + (2/3)k

Formula used

â=a/|a|

Textbook formal language

Divide by magnitude 9.

Easy language (same calculation)

3,6,6 divide by 9.

Why this formula

Magnitude of â is 1.

Exam tip

Keep signs.

Common mistakes

(3,−6,6)
Dividing by 3 only

Question 6 of 10Addition

If a=i+j and b=i−j, find a+b and |a+b|.

Parallelogram law

1) a+b=2i; |a+b|=2.

Answer: a+b=2i; |a+b|=2

Formula used

Parallelogram law

Textbook formal language

Add components; magnitude of resulting vector.

Easy language (same calculation)

j cancels; left 2i.

Why this formula

a and b are equal length √2 and orthogonal-ish; sum along x.

Exam tip

Do not multiply vectors.

Common mistakes

Question 7 of 10Magnitude

Find |3i − 6j + 6k|.

|a|=√(a₁²+a₂²+a₃²)

1) √(9+36+36)=√81=9.

Answer: 9

Formula used

|a|=√(a₁²+a₂²+a₃²)

Textbook formal language

Euclidean norm of the vector.

Easy language (same calculation)

3-6-6 scales a 1-2-2 unit pattern times 3.

Why this formula

Unit vector would be (1/3,−2/3,2/3).

Exam tip

Check each algebraic step carefully.

Common mistakes

√81 left unsimplified
3

Question 8 of 10Dot

Compute <1, 2, 3> · <4, −5, 6>.

a·b = |a||b|cosθ

1) 1·4 + 2·(−5) + 3·6 = 4 − 10 + 18 = 12.

Answer: 12

Formula used

a·b = |a||b|cosθ

Textbook formal language

Sum of component products.

Easy language (same calculation)

Scalar result.

Why this formula

0 would mean perpendicular.

Exam tip

Check each algebraic step carefully.

Common mistakes

Vector answer
−12

Question 9 of 10Perp

Are a=<1,2> and b=<2,−1> perpendicular?

a·b=0

1) a·b = 2 − 2 = 0 ⇒ yes, perpendicular.

Answer: Yes

Formula used

a·b=0

Textbook formal language

Zero dot product (in 2D or 3D).

Easy language (same calculation)

Slopes 2 and −1/2 also show perpendicular lines in plane.

Why this formula

Non-zero vectors required for geometric ⊥.

Exam tip

Check each algebraic step carefully.

Common mistakes

No
Parallel

Question 10 of 10Cross

Find i × j.

i×j=k

1) By right-hand rule / cyclic order, i×j = k.

Answer: k

Formula used

i×j=k

Textbook formal language

Standard basis cross products.

Easy language (same calculation)

i→j→k→i cyclic.

Why this formula

j×i = −k.

Exam tip

Check each algebraic step carefully.