How to Use This Page
This page contains complete exercise solutions for Chapter 6 — Vector Algebra from the Samacheer Kalvi 11th Business Maths textbook (Tamil Nadu State Board, English Medium, Commerce stream).
Work through each section in order:
1. Key Concepts — quick revision tables before you start solving. 2. MCQs, Fill in the Blanks, True or False, Match the Following — one-mark practice. 3. Answer Briefly — two-mark short answers. 4. Answer in Detail — five-mark problems with fully worked numerical examples. 5. Common Mistakes and Exam Checklist — final review before your exam.
Every answer follows the TN Board marking scheme. Code blocks show step-by-step calculations so you can follow the working.
Key Concepts (Quick Recap)
Scalar vs Vector
| Property | Scalar | Vector |
|---|---|---|
| Definition | Has only magnitude | Has both magnitude and direction |
| Examples | Mass, temperature, speed | Displacement, velocity, force |
| Notation | Ordinary letter (m, t) | Bold (a) or with arrow (a⃗) |
| Addition | Simple arithmetic | Triangle or parallelogram law |
| Representation | Single number | Components along axes |
Types of Vectors
| Type | Definition | ||
|---|---|---|---|
| Zero vector (0⃗) | Magnitude = 0; no specific direction | ||
| Unit vector | Magnitude = 1; denoted â = a/ | a | |
| Equal vectors | Same magnitude and same direction | ||
| Negative vector | Same magnitude, opposite direction | ||
| Collinear vectors | Parallel to the same line (may differ in magnitude) | ||
| Coplanar vectors | Lie in the same plane | ||
| Position vector | Vector from origin O to point P; OP⃗ = xî + yĵ + zk̂ |
Vector Addition
| Law | Statement |
|---|---|
| Triangle law | If two sides of a triangle represent two vectors, the third side (from tail of first to head of second) gives the resultant: a⃗ + b⃗ |
| Parallelogram law | If two vectors are drawn from the same point as adjacent sides of a parallelogram, the diagonal from that point gives the resultant |
Scalar Multiplication
Magnitude of a Vector
Dot (Scalar) Product
Cross (Vector) Product
Direction Cosines and Ratios
Projection of a Vector
Angle Between Two Vectors
I. Choose the Best Answer (MCQs)
| 1. If a⃗ = 2î + 3ĵ − k̂, then | a⃗ | is: |
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2. If a⃗ · b⃗ = 0, then the angle between a⃗ and b⃗ is:
3. The direction cosines of the vector î + ĵ + k̂ are:
4. If a⃗ × b⃗ = 0⃗, then:
5. The unit vector in the direction of a⃗ = 3î − 4ĵ + 12k̂ is:
6. î × ĵ equals:
7. The projection of a⃗ on b⃗ is:
| **Answer: (b) (a⃗ · b⃗) / | b⃗ | ** |
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II. Fill in the Blanks
| 1. If a⃗ = a₁î + a₂ĵ + a₃k̂, then | a⃗ | = √(a₁² + a₂² + a₃²) |
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2. The direction cosines l, m, n satisfy the relation l² + m² + n² = 1
3. If a⃗ ⊥ b⃗, then a⃗ · b⃗ = 0
4. a⃗ × b⃗ = −(b⃗ × a⃗) (anti-commutative property)
| 5. The unit vector in the direction of a⃗ is **a⃗/ | a⃗ | ** |
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6. If a⃗ = 2î − ĵ + 3k̂, then 2a⃗ = 4î − 2ĵ + 6k̂
| 7. The dot product of parallel vectors equals ** | a⃗ | b⃗ | ** (since cos 0° = 1) |
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8. The cross product of two parallel vectors is 0⃗ (zero vector)
9. If the position vector of point P is 3î − 2ĵ + 5k̂, then the coordinates of P are (3, −2, 5)
10. î · k̂ = 0 (since î and k̂ are perpendicular)
11. The magnitude of the vector −6î + 8ĵ is 10
12. If l = 1/2 and m = 1/2 are direction cosines, then n = 1/√2
III. True or False
1. "The dot product of two vectors is always a vector."
| False. The dot product a⃗ · b⃗ = | a⃗ | b⃗ | cos θ is always a scalar quantity, not a vector. |
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2. "If the direction cosines of a vector are l, m, n, then l² + m² + n² = 1."
True. This is the fundamental relation satisfied by direction cosines.
3. "Two vectors are perpendicular if their dot product equals 1."
False. Two non-zero vectors are perpendicular if their dot product equals 0, not 1. The correction: a⃗ · b⃗ = 0 ⟺ a⃗ ⊥ b⃗.
4. "The cross product of a vector with itself is the zero vector."
True. a⃗ × a⃗ = 0⃗ because the angle between a⃗ and itself is 0°, and sin 0° = 0.
5. "The magnitude of a vector is always negative."
| False. The magnitude | a⃗ | = √(a₁² + a₂² + a₃²) is always non-negative (≥ 0). It is zero only for the zero vector. |
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IV. Match the Following
| Column A | Column B | ||
|---|---|---|---|
| 1. a⃗ · b⃗ = 0 | (c) Vectors are perpendicular | ||
| 2. a⃗ × b⃗ = 0⃗ | (e) Vectors are parallel | ||
| 3. a⃗ · a⃗ | (a) \ | a⃗\ | ² |
| 4. l² + m² + n² | (d) 1 | ||
| 5. a⃗/ | a⃗ | (f) Unit vector | |
| 6. î × k̂ | (b) −ĵ |
Explanations:
1. a⃗ · b⃗ = 0 ⟹ cos θ = 0 ⟹ θ = 90°, so vectors are perpendicular. 2. a⃗ × b⃗ = 0⃗ ⟹ sin θ = 0 ⟹ θ = 0° or 180°, so vectors are parallel.
4. Direction cosines always satisfy l² + m² + n² = 1. 5. Dividing a vector by its magnitude gives a unit vector. 6. î × k̂ = −(k̂ × î) = −ĵ.
| 3. a⃗ · a⃗ = | a⃗ | ² cos 0° = | a⃗ | ². |
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V. Answer Briefly
1. Distinguish between scalar and vector quantities with two examples each.
| Scalar | Vector | |
|---|---|---|
| Definition | Has only magnitude | Has both magnitude and direction |
| Example 1 | Mass (5 kg) | Displacement (5 m east) |
| Example 2 | Temperature (37 °C) | Velocity (20 m/s north) |
A scalar is fully described by its magnitude alone. A vector requires both magnitude and direction for complete specification. Scalars follow ordinary algebra; vectors follow vector algebra (triangle/parallelogram laws).
2. Write the properties of the dot product.
3. What are direction cosines and direction ratios? How are they related?
Direction cosines: If a vector a⃗ makes angles α, β, γ with the positive x, y, z axes respectively, then cos α, cos β, cos γ are called direction cosines, denoted l, m, n. They satisfy:
Direction ratios: Three numbers a, b, c proportional to the direction cosines are called direction ratios. The relation between them:
Direction ratios are not unique (any proportional set works), but direction cosines are unique (they satisfy l² + m² + n² = 1).
4. Find the direction cosines of the vector 2î + 3ĵ + 6k̂.
Direction cosines: 2/7, 3/7, 6/7
VI. Answer in Detail
1. Vector Addition and Magnitude Calculation
Explain the triangle law and parallelogram law of vector addition. Find the magnitude and direction cosines of the resultant vector when a⃗ = î + 2ĵ + 3k̂ and b⃗ = 4î − 5ĵ + 6k̂.
Triangle Law of Vector Addition:
If two vectors are represented as two sides of a triangle taken in order, then the third side taken in the opposite order represents the resultant vector. If OA⃗ = a⃗ and AB⃗ = b⃗, then OB⃗ = a⃗ + b⃗.
Parallelogram Law of Vector Addition:
If two vectors are represented as adjacent sides of a parallelogram drawn from the same point, then the diagonal from that point represents the resultant vector. If OA⃗ = a⃗ and OB⃗ = b⃗, then OC⃗ (diagonal) = a⃗ + b⃗.
Worked Example:
2. Dot Product and Angle Between Vectors
Explain the scalar (dot) product of two vectors and its properties. Find the angle between the vectors a⃗ = î + 2ĵ + 3k̂ and b⃗ = 2î − ĵ + 4k̂.
Definition: The dot product of two vectors a⃗ and b⃗ is defined as:
Properties:
Worked Example:
The angle between a⃗ and b⃗ is cos⁻¹(2√6/7) ≈ 45.6°
3. Cross Product and Its Applications
Explain the vector (cross) product of two vectors and its properties. Find the cross product and the area of the parallelogram formed by a⃗ = 3î − ĵ + 2k̂ and b⃗ = î + 2ĵ − k̂.
Definition: The cross product of two vectors a⃗ and b⃗ is defined as:
Determinant form:
Properties:
Worked Example:
Cross product: −3î + 5ĵ + 7k̂; Area of parallelogram: √83 sq. units
Common Mistakes to Avoid
1. Confusing dot and cross products — The dot product gives a scalar; the cross product gives a vector. Never write a⃗ · b⃗ as a vector or a⃗ × b⃗ as a scalar.
2. Forgetting the negative sign in cross product determinant — When expanding the determinant, the ĵ component has a minus sign: −ĵ(a₁b₃ − a₃b₁). Many students forget this.
3. Applying dot product rules to cross product — The cross product is anti-commutative (a⃗ × b⃗ = −b⃗ × a⃗), not commutative. This is different from the dot product.
4. Direction cosines vs direction ratios — Direction cosines satisfy l² + m² + n² = 1; direction ratios do not. Do not assume a² + b² + c² = 1 for direction ratios.
| 5. Magnitude must be positive — | a⃗ | = √(a₁² + a₂² + a₃²) is always ≥ 0. Never write a negative magnitude. |
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| 6. Unit vector formula — â = a⃗/ | a⃗ | . Some students write â = | a⃗ | /a⃗ or forget to divide each component by the magnitude. |
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| 7. Angle between vectors — Use cos θ = (a⃗ · b⃗)/( | a⃗ | b⃗ | ), NOT sin θ with the dot product. The sine formula uses the cross product: sin θ = | a⃗ × b⃗ | /( | a⃗ | b⃗ | ). |
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8. Zero vector vs scalar zero — The zero vector 0⃗ = 0î + 0ĵ + 0k̂ is a vector, not the scalar 0. Writing just "0" in vector equations can cause confusion.
Exam Checklist
Use this checklist before your exam. Tick each item once you are confident.
- [ ] I can distinguish between scalars and vectors with examples
- [ ] I know the types of vectors: zero, unit, equal, negative, collinear, coplanar
- [ ] I can find the position vector of a point and write components
- [ ] I can add vectors using both triangle and parallelogram laws
- [ ] I can compute the magnitude |a⃗| = √(a₁² + a₂² + a₃²)
- [ ] I can compute the dot product a⃗ · b⃗ = a₁b₁ + a₂b₂ + a₃b₃ and use it to find angles
- [ ] I can compute the cross product using the determinant method
- [ ] I know that |a⃗ × b⃗| gives the area of a parallelogram and (1/2)|a⃗ × b⃗| gives the area of a triangle
- [ ] I can find direction cosines from a vector and verify l² + m² + n² = 1
- [ ] I can find the projection of one vector on another: proj = (a⃗ · b⃗)/|b⃗|