Class 12 Mathematics Notes Chapter 10 (Vector Algebra) – Examplar Problems (English) Book

Alright class, let's get started with a very important chapter for your competitive exams – Vector Algebra. Understanding vectors is crucial not just for mathematics, but also for physics and engineering applications. We'll cover the core concepts as presented in your NCERT syllabus and Exemplar problems, focusing on what you need for exam success.

Chapter 10: Vector Algebra - Detailed Notes for Government Exam Preparation

1. Introduction: Scalars and Vectors

• Scalar: A quantity that has only magnitude. Examples: Length, mass, time, temperature, distance, speed, work, energy.
• Vector: A quantity that has both magnitude (or modulus) and direction. Examples: Displacement, velocity, acceleration, force, momentum, electric field.
• Representation: A vector is represented by a directed line segment. If a vector goes from point A (initial point) to point B (terminal point), it's denoted as vec(AB) or by a single bold letter like a or vec(a).
• Magnitude: The magnitude (or length or modulus) of vector vec(AB) is the length of the line segment AB and is denoted by |vec(AB)| or |**a**| or a. Magnitude is always non-negative.

2. Types of Vectors

• Zero Vector (Null Vector): A vector whose initial and terminal points coincide. Its magnitude is 0, and it has no specific direction (or can be regarded as having any direction). Denoted by vec(0) or 0.
• Unit Vector: A vector whose magnitude is exactly 1. A unit vector in the direction of a given vector a is denoted by â (read as 'a cap') and is calculated as: â = **a** / |**a**|.
  • Standard unit vectors along the positive x, y, and z axes are denoted by i, j, and k respectively. |**i**| = |**j**| = |**k**| = 1.
• Coinitial Vectors: Two or more vectors having the same initial point.
• Collinear Vectors: Two or more vectors are collinear if they are parallel to the same line, irrespective of their magnitudes and directions. If a and b are collinear, then a = λb for some non-zero scalar λ.
• Equal Vectors: Two vectors a and b are equal if they have the same magnitude and the same direction, regardless of their initial points. If a = b, then |**a**| = |**b**| and they have the same direction.
• Negative of a Vector: A vector having the same magnitude as a given vector a but the opposite direction. Denoted by -a. vec(BA) = -vec(AB).

3. Addition of Vectors

• Triangle Law: If two vectors are represented in magnitude and direction by two sides of a triangle taken in order, then their sum (resultant) is represented in magnitude and direction by the third side of the triangle taken in the opposite order. If vec(AB) = **a** and vec(BC) = **b**, then vec(AC) = **a** + **b**.
• Parallelogram Law: If two vectors are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, then their sum (resultant) is represented in magnitude and direction by the diagonal of the parallelogram passing through that same point. If vec(OA) = **a** and vec(OB) = **b**, then vec(OC) = **a** + **b**, where OACB is a parallelogram.
• Properties of Vector Addition:
  • Commutative: a + b = b + a
  • Associative: (a + b) + c = a + (b + c)
  • Additive Identity: a + 0 = 0 + a = a (0 is the zero vector)
  • Additive Inverse: a + (-a) = 0 (-a is the negative of a)

4. Multiplication of a Vector by a Scalar

• If a is a vector and λ is a scalar, then λa is a vector whose magnitude is |λ| |**a**|.
• The direction of λa is the same as a if λ > 0, and opposite to a if λ < 0. If λ = 0, then λa = 0.
• Properties:
  • λ(a + b) = λa + λb
  • (λ + μ)a = λa + μa
  • λ(μa) = (λμ)a
• Components of a Vector: Any vector r in 3D space can be represented as a linear combination of the orthogonal unit vectors i, j, k:
  **r** = x**i** + y**j** + z**k
  Here, x, y, z are the scalar components of r along the x, y, z axes, respectively. x**i**, y**j**, z**k are the vector components.
• Magnitude in Component Form: |**r**| = sqrt(x² + y² + z²)
• Addition/Subtraction in Component Form: If **a** = a₁**i** + a₂**j** + a₃**k and **b** = b₁**i** + b₂**j** + b₃**k, then
  **a** + **b** = (a₁ + b₁)**i** + (a₂ + b₂)**j** + (a₃ + b₃)**k
**a** - **b** = (a₁ - b₁)**i** + (a₂ - b₂)**j** + (a₃ - b₃)**k
• Scalar Multiplication in Component Form: λ**a** = (λa₁)**i** + (λa₂)**j** + (λa₃)**k
• Condition for Collinearity (Component Form): Two vectors a and b are collinear if and only if b₁/a₁ = b₂/a₂ = b₃/a₃ = λ (provided a₁, a₂, a₃ ≠ 0).
• Direction Cosines (l, m, n): If a vector r makes angles α, β, γ with the positive x, y, z axes respectively, then cos α, cos β, cos γ are its direction cosines (DCs).
  l = cos α = x / |**r**|, m = cos β = y / |**r**|, n = cos γ = z / |**r**|
  Important Property: l² + m² + n² = 1 (i.e., cos²α + cos²β + cos²γ = 1)
• Direction Ratios (a, b, c): Any three numbers proportional to the direction cosines are called direction ratios (DRs). If a, b, c are DRs, then l = ± a/√(a²+b²+c²), m = ± b/√(a²+b²+c²), n = ± c/√(a²+b²+c²). The scalar components (x, y, z) of a vector **r** = x**i** + y**j** + z**k are its direction ratios.

5. Position Vector (PV)

• The position vector of a point P with respect to an origin O is the vector vec(OP). If P has coordinates (x, y, z), then vec(OP) = x**i** + y**j** + z**k.
• Vector Joining Two Points: If P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) are two points, the vector joining P₁ to P₂ is:
  vec(P₁P₂) = PV of P₂ - PV of P₁
vec(P₁P₂) = (x₂ - x₁)**i** + (y₂ - y₁)**j** + (z₂ - z₁)**k
• Magnitude: |vec(P₁P₂)| = sqrt((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²) (This is just the distance formula).

6. Section Formula

Let A and B be two points with position vectors a and b respectively. Let R be a point dividing the line segment AB.

• Internal Division: If R divides AB internally in the ratio m:n, its position vector r is given by:
  **r** = (m**b** + n**a**) / (m + n)
• External Division: If R divides AB externally in the ratio m:n, its position vector r is given by:
  **r** = (m**b** - n**a**) / (m - n)
• Mid-point Formula: If R is the midpoint of AB (ratio 1:1), then:
  **r** = (**a** + **b**) / 2

7. Product of Two Vectors

• Scalar (Dot) Product:

  • Definition: **a** · **b** = |**a**| |**b**| cos θ, where θ is the angle between a and b (0 ≤ θ ≤ π).
  • The result is a scalar.
  • Geometric Interpretation: **a** · **b** is the product of the magnitude of a and the projection of b onto a (or vice versa).
  • Properties:
    • **a** · **b** = **b** · **a** (Commutative)
    • **a** · (**b** + **c**) = **a** · **b** + **a** · **c** (Distributive over addition)
    • (λ**a**) · **b** = λ(**a** · **b**) = **a** · (λ**b**)
    • **a** · **a** = |**a**|² (Important: |**a**| = sqrt(**a** · **a**))
    • **a** · **b** = 0 <=> a is perpendicular to b (provided a, b ≠ 0)
    • Angle between a and b: cos θ = (**a** · **b**) / (|**a**| |**b**|)
    • For orthogonal unit vectors: **i** · **i** = **j** · **j** = **k** · **k** = 1 and **i** · **j** = **j** · **k** = **k** · **i** = 0.
  • Component Form: If **a** = a₁**i** + a₂**j** + a₃**k and **b** = b₁**i** + b₂**j** + b₃**k, then:
    **a** · **b** = a₁b₁ + a₂b₂ + a₃b₃
  • Projection:
    • Projection of a on b (scalar): (**a** · **b**) / |**b**|
    • Projection vector of a on b: ((**a** · **b**) / |**b**|²) **b**

• Vector (Cross) Product:

  • Definition: **a** × **b** = |**a**| |**b**| sin θ n̂, where θ is the angle between a and b (0 ≤ θ ≤ π), and n̂ is a unit vector perpendicular to the plane containing a and b, such that a, b, n̂ form a right-handed system.
  • The result is a vector.
  • Geometric Interpretation: |**a** × **b**| represents the area of the parallelogram with adjacent sides a and b.
  • Properties:
    • **a** × **b** = - (**b** × **a**) (Not commutative)
    • **a** × (**b** + **c**) = **a** × **b** + **a** × **c** (Distributive over addition)
    • (λ**a**) × **b** = λ(**a** × **b**) = **a** × (λ**b**)
    • **a** × **a** = **0**
    • **a** × **b** = **0** <=> a is parallel (collinear) to b (provided a, b ≠ 0)
    • Angle between a and b: sin θ = |**a** × **b**| / (|**a**| |**b**|)
    • For orthogonal unit vectors: **i** × **i** = **j** × **j** = **k** × **k** = **0**
    • **i** × **j** = **k**, **j** × **k** = **i**, **k** × **i** = **j**
    • **j** × **i** = -**k**, **k** × **j** = -**i**, **i** × **k** = -**j**
  • Component Form: If **a** = a₁**i** + a₂**j** + a₃**k and **b** = b₁**i** + b₂**j** + b₃**k, then:
    **a** × **b** = | **i** **j** **k** |
| a₁ a₂ a₃ |
| b₁ b₂ b₃ |
= (a₂b₃ - a₃b₂) **i** - (a₁b₃ - a₃b₁) **j** + (a₁b₂ - a₂b₁) **k
  • Area of Parallelogram (adjacent sides a, b): Area = |**a** × **b**|
  • Area of Triangle (adjacent sides a, b): Area = (1/2) |**a** × **b**|
  • Area of Triangle (vertices A, B, C with PVs a, b, c): Area = (1/2) |vec(AB) × vec(AC)| = (1/2) |(**b** - **a**) × (**c** - **a**)|

8. Scalar Triple Product (STP)

• Definition: Given three vectors a, b, c, their scalar triple product is defined as (**a** × **b**) · **c**.
• Notation: [**a** **b** **c**] or (**a**, **b**, **c**).
• Geometric Interpretation: | [**a** **b** **c**] | represents the volume of the parallelepiped whose coterminous edges are a, b, c.
• Properties:
  • ( **a** × **b** ) · **c** = **a** · ( **b** × **c** ) (Dot and cross can be interchanged)
  • [**a** **b** **c**] = [**b** **c** **a**] = [**c** **a** **b**] (Cyclic permutation retains value)
  • [**a** **b** **c**] = -[**a** **c** **b**] (Interchanging any two vectors negates the value)
  • [**a** **a** **b**] = 0 (If any two vectors are equal or collinear, STP is 0)
  • Condition for Coplanarity: Three vectors a, b, c are coplanar if and only if [**a** **b** **c**] = 0 (provided they are non-zero). This means the volume of the parallelepiped formed by them is zero.
• Component Form: If **a** = a₁**i** + a₂**j** + a₃**k, **b** = b₁**i** + b₂**j** + b₃**k, **c** = c₁**i** + c₂**j** + c₃**k, then:
  [**a** **b** **c**] = | a₁ a₂ a₃ |
| b₁ b₂ b₃ |
| c₁ c₂ c₃ |
• Volume of Tetrahedron with coterminous edges a, b, c: Volume = (1/6) |[**a** **b** **c**]|
• Volume of Tetrahedron with vertices A, B, C, D: Volume = (1/6) |[vec(AB) vec(AC) vec(AD)]|

9. Vector Triple Product (VTP)

• Definition: **a** × (**b** × **c**)
• Result: A vector lying in the plane of b and c, and perpendicular to a.
• Expansion Formula: **a** × (**b** × **c**) = (**a** · **c**) **b** - (**a** · **b**) **c** (Remember as "ACB - ABC" dot products)
• Note: (**a** × **b**) × **c** = (**a** · **c**) **b** - (**b** · **c**) **a**. VTP is not associative.

Key Formulas Summary:

• Unit Vector: â = **a** / |**a**|
• Component Magnitude: |x**i** + y**j** + z**k| = sqrt(x² + y² + z²)
• DCs Property: l² + m² + n² = 1
• Section Formula (Internal): **r** = (m**b** + n**a**) / (m + n)
• Dot Product: **a** · **b** = |**a**| |**b**| cos θ = a₁b₁ + a₂b₂ + a₃b₃
• Condition for Perpendicularity: **a** · **b** = 0
• Projection of a on b: (**a** · **b**) / |**b**|
• Cross Product: **a** × **b** = |**a**| |**b**| sin θ n̂ = | **i** **j** **k** | ... (determinant)
• Condition for Parallelism: **a** × **b** = **0**
• Area of Parallelogram: |**a** × **b**|
• Area of Triangle: (1/2) |**a** × **b**|
• Scalar Triple Product: [**a** **b** **c**] = (**a** × **b**) · **c** = | a₁ a₂ a₃ | ... (determinant)
• Condition for Coplanarity: [**a** **b** **c**] = 0
• Volume of Parallelepiped: | [**a** **b** **c**] |
• Vector Triple Product: **a** × (**b** × **c**) = (**a** · **c**) **b** - (**a** · **b**) **c**

Problem-Solving Tips for Exams:

• Visualize vectors whenever possible (arrows in 2D or 3D space).
• Clearly distinguish between scalar and vector quantities in results.
• Remember the conditions for perpendicularity (**a** · **b** = 0) and parallelism (**a** × **b** = **0** or **a** = λ**b**).
• Use component form for most numerical calculations involving dot, cross, and scalar triple products.
• Pay attention to the wording: "projection" vs "projection vector", "magnitude", "direction cosines/ratios".
• Practice problems involving geometric applications (area, volume, collinearity, coplanarity).
• Know the properties of dot, cross, and scalar triple products thoroughly.

Multiple Choice Questions (MCQs)

1. If a and b are two collinear vectors, then which of the following are incorrect:
   (A) b = λa, for some scalar λ
   (B) a = ±b
   (C) the respective components of a and b are proportional
   (D) both the vectors a and b have same direction, but different magnitudes.

2. Let a and b be two unit vectors and θ is the angle between them. Then a + b is a unit vector if:
   (A) θ = π/4
   (B) θ = π/3
   (C) θ = π/2
   (D) θ = 2π/3

3. The vector in the direction of the vector i - 2j + 2k that has magnitude 9 is:
   (A) i - 2j + 2k
   (B) (i - 2j + 2k) / 3
   (C) 3(i - 2j + 2k)
   (D) 9(i - 2j + 2k)

4. The projection of the vector a = 2i + 3j + 2k on the vector b = i + 2j + k is:
   (A) 10 / √6
   (B) 10 / √17
   (C) 8 / √6
   (D) 8 / √17

5. If a, b, c are three vectors such that a + b + c = 0 and |a|=3, |b|=5, |c|=7, then the angle between a and b is:
   (A) π/6
   (B) π/3
   (C) 2π/3
   (D) π/2

6. The area of the parallelogram whose adjacent sides are determined by the vectors a = i - j + 3k and b = 2i - 7j + k is:
   (A) 15√2
   (B) √450
   (C) 10√3
   (D) 20

7. If the vectors a = 2i - j + k, b = i + 2j - 3k and c = 3i + pj + 5k are coplanar, then the value of p is:
   (A) -2
   (B) 2
   (C) -4
   (D) 4

8. The value of i · (j × k) + j · (i × k) + k · (i × j) is:
   (A) 0
   (B) -1
   (C) 1
   (D) 3

9. If |a| = 10, |b| = 2 and a · b = 12, then the value of |a × b| is:
   (A) 5
   (B) 10
   (C) 12
   (D) 16

10. The position vectors of points A, B, C are a, b, c respectively. The points A, B, C are collinear if:
    (A) a × b + b × c + c × a = 0
    (B) a · b + b · c + c · a = 0
    (C) [a b c] = 0
    (D) a + b + c = 0

Answers to MCQs:

1. (D) Collinear vectors can have opposite directions, and they can also have the same magnitude (e.g., a and a). Option D states they must have the same direction and different magnitudes, which is incorrect.
2. (D) |a + b| = 1. Squaring both sides: |a + b|² = 1 => (a + b) · (a + b) = 1 => |a|² + |b|² + 2(a · b) = 1. Since |a|=|b|=1, we get 1 + 1 + 2(1)(1)cos θ = 1 => 2 + 2cos θ = 1 => 2cos θ = -1 => cos θ = -1/2. Hence, θ = 2π/3.
3. (C) Let v = i - 2j + 2k. |v| = √(1² + (-2)² + 2²) = √(1 + 4 + 4) = √9 = 3. Unit vector in direction of v is v̂ = v / |v| = (i - 2j + 2k) / 3. Vector with magnitude 9 in this direction is 9v̂ = 9 * [(i - 2j + 2k) / 3] = 3(i - 2j + 2k).
4. (A) Projection of a on b = (a · b) / |b|.
   a · b = (2)(1) + (3)(2) + (2)(1) = 2 + 6 + 2 = 10.
   |b| = √(1² + 2² + 1²) = √(1 + 4 + 1) = √6.
   Projection = 10 / √6.
5. (B) a + b + c = 0 => a + b = -c. Squaring both sides: |a + b|² = |-c|² => |a|² + |b|² + 2a · b = |c|² => |a|² + |b|² + 2|a||b|cos θ = |c|².
   3² + 5² + 2(3)(5)cos θ = 7² => 9 + 25 + 30 cos θ = 49 => 34 + 30 cos θ = 49 => 30 cos θ = 15 => cos θ = 1/2. Hence, θ = π/3.
6. (A) Area = |a × b|.
   a × b = | i j k | = i(-1 - (-21)) - j(1 - 6) + k(-7 - (-2)) = 20i + 5j - 5k.
   |a × b| = √(20² + 5² + (-5)²) = √(400 + 25 + 25) = √450 = √(225 * 2) = 15√2. (Note: B is numerically correct but A is the simplified form). Let's stick with A.
7. (C) Vectors are coplanar if [a b c] = 0.
   | 2 -1 1 |
   | 1 2 -3 | = 0
   | 3 p 5 |
   2(10 - (-3p)) - (-1)(5 - (-9)) + 1(p - 6) = 0
   2(10 + 3p) + 1(14) + (p - 6) = 0
   20 + 6p + 14 + p - 6 = 0
   7p + 28 = 0 => 7p = -28 => p = -4.
8. (C) i · (j × k) = i · i = 1.
   j · (i × k) = j · (-j) = -1.
   k · (i × j) = k · k = 1.
   Sum = 1 + (-1) + 1 = 1. (Alternatively, recognize these are [i j k], [j i k], [k i j]. [i j k]=1, [j i k]=-[i j k]=-1, [k i j]=[i j k]=1. Sum = 1 - 1 + 1 = 1).
9. (D) We know |a × b|² + (a · b)² = |a|² |b|².
   |a × b|² + (12)² = (10)² (2)²
   |a × b|² + 144 = 100 * 4 = 400
   |a × b|² = 400 - 144 = 256
   |a × b| = √256 = 16.
10. (A) Points A, B, C are collinear if vectors vec(AB) and vec(BC) are collinear.
    vec(AB) = **b** - **a
vec(BC) = **c** - **b
    Collinear means vec(AB) × vec(BC) = **0**.
    (b - a) × (c - b) = 0
    b × c - b × b - a × c + a × b = 0
    b × c - 0 + c × a + a × b = 0 (since -a × c = c × a)
    a × b + b × c + c × a = 0. (This condition is also equivalent to the area of the triangle ABC being zero).

Make sure you thoroughly understand these concepts and practice solving problems from your NCERT book and the Exemplar. Good luck!