Vector Algebra Class 12 Maths Notes & Mind Map

01 · Explore

Basic Concepts and Definitions

In physics and mathematics, quantities are broadly classified into scalars and vectors based on whether they possess direction.

A scalar quantity is one that involves only magnitude, represented by a real number. Examples include mass, time, distance, and temperature. For instance, a height of 1.6 meters is a scalar because it does not require a direction to be fully understood.

A vector quantity possesses both magnitude and direction. Examples include displacement, velocity, acceleration, and force. Geometrically, a vector is represented by a directed line segment. For a vector AB, point A is the initial point and point B is the terminal point. The magnitude of the vector, denoted as |AB|, is the length of this segment.

A position vector describes the location of a point P(x, y, z) relative to the origin O(0, 0, 0). It is denoted as r = xî + yĵ + zk̂. The magnitude of this position vector is calculated using the distance formula: |r| = √(x² + y² + z²).

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Question

What is the magnitude of a position vector for point P(3, -4, 0)?

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NCERT reference: chapter PDF pages 1, 2.

02 · Explore

Direction Cosines and Ratios

Direction cosines and ratios provide a mathematical way to describe the orientation of a vector in 3D space.

Consider a position vector r making angles α, β, and γ with the positive x, y, and z axes respectively. These are called direction angles. The cosines of these angles, l = cos α, m = cos β, and n = cos γ, are known as the direction cosines (DCs) of the vector.

Direction ratios (DRs) are any numbers a, b, and c that are proportional to the direction cosines. For a vector r = xî + yĵ + zk̂, the scalar components x, y, and z serve as the direction ratios. A key property of direction cosines is that l² + m² + n² = 1, whereas for direction ratios, a² + b² + c² is generally not equal to 1.

Term	Symbol	Relation to Magnitude (r)
Direction Cosine l	cos α	x / r
Direction Cosine m	cos β	y / r
Direction Cosine n	cos γ	z / r

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Question

If a vector has direction ratios (1, 1, √2), find its direction cosines.

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NCERT reference: chapter PDF pages 3, 4.

03 · Explore

Types of Vectors

Vectors are categorized based on their magnitude, direction, and starting points.

A Zero Vector (or null vector) has zero magnitude and its initial and terminal points coincide. It is denoted as 0⃑. A Unit Vector has a magnitude of exactly 1 unit. The unit vector in the direction of vector a is denoted as â and is calculated as a⃑ / |a⃑|.

Coinitial vectors share the same starting point. Collinear vectors are parallel to the same line regardless of their magnitudes. Equal vectors have the same magnitude and direction, even if they start at different positions. The Negative of a vector has the same magnitude but the exact opposite direction.

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Question

True or False: Two collinear vectors must always have the same magnitude.

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NCERT reference: chapter PDF pages 4, 5.

04 · Explore

Addition of Vectors

Vector addition follows specific geometric laws rather than simple algebraic summation.

The Triangle Law of Vector Addition states that if two vectors are represented by two sides of a triangle taken in order, their sum (resultant) is represented by the third side of the triangle taken in the opposite order. Mathematically, AB⃑ + BC⃑ = AC⃑.

The Parallelogram Law of Vector Addition states that if two vectors are represented by the adjacent sides of a parallelogram, their sum is represented by the diagonal passing through their common initial point. These two laws are equivalent.

Vector addition is commutative (a⃑ + b⃑ = b⃑ + a⃑) and associative (a⃑ + (b⃑ + c⃑) = (a⃑ + b⃑) + c⃑). The zero vector acts as the additive identity (a⃑ + 0⃑ = a⃑).

Triangle Law Sequence

1
Vector A to B
Position the first vector from point A to point B.
2
Vector B to C
Place the initial point of the second vector at the terminal point (B) of the first.
3
Resultant A to C
The sum is the vector directed from the start of the first (A) to the end of the second (C).

The geometric process of adding two vectors using the triangle law.

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Question

In a triangle ABC, what is the sum of vectors AB⃑ + BC⃑ + CA⃑?

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NCERT reference: chapter PDF pages 6, 7, 8, 9.

05 · Explore

Components of a Vector

Expressing vectors in component form allows for easy algebraic manipulation.

In a 3D coordinate system, î, ĵ, and k̂ are unit vectors along the x, y, and z axes. Any vector r can be written as r = xî + yĵ + zk̂, where x, y, and z are scalar components.

To add vectors in component form, simply add their corresponding components: (a₁î + a₂ĵ + a₃k̂) + (b₁î + b₂ĵ + b₃k̂) = (a₁+b₁)î + (a₂+b₂)ĵ + (a₃+b₃)k̂. Scalar multiplication distributes over the components: λ(xî + yĵ + zk̂) = (λx)î + (λy)ĵ + (λz)k̂.

Finding a Unit Vector

a⃑ = 2î + 3ĵ + k̂

1. Calculate magnitude: |a⃑| = √(2² + 3² + 1²) = √(4 + 9 + 1) = √14. 2. Divide vector by magnitude: â = (2/√14)î + (3/√14)ĵ + (1/√14)k̂.

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Question

Find the values of x and y if xî + 2ĵ = 3î + yĵ.

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NCERT reference: chapter PDF pages 10, 11, 12.

06 · Explore

Vector Joining Two Points

The vector between two points is found by subtracting the position vector of the initial point from the terminal point.

If P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) are two points, the vector P₁P₂⃑ is given by (x₂ - x₁)î + (y₂ - y₁)ĵ + (z₂ - z₁)k̂.

The magnitude of this vector is |P₁P₂⃑| = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]. This is essentially the distance formula in 3D space.

Vector Calculation

P(1, 2, 3), Q(4, 5, 6) → PQ⃑

PQ⃑ = (4-1)î + (5-2)ĵ + (6-3)k̂ = 3î + 3ĵ + 3k̂. Magnitude |PQ⃑| = √(3² + 3² + 3²) = √27 = 3√3.

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Question

Find the vector directed from A(2, 3, 0) to B(-1, -2, -4).

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NCERT reference: chapter PDF pages 14, 15.

07 · Explore

Section Formula

The section formula determines the position vector of a point that divides a line segment in a given ratio.

Internal Division: If a point R divides the line segment joining points P (position vector a⃑) and Q (position vector b⃑) internally in the ratio m:n, its position vector r⃑ is given by r⃑ = (mb⃑ + na⃑) / (m + n).

External Division: If point R divides the segment externally in the ratio m:n, the position vector is r⃑ = (mb⃑ - na⃑) / (m - n).

Midpoint: If R is the midpoint, the ratio is 1:1, and the formula simplifies to r⃑ = (a⃑ + b⃑) / 2.

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Question

Find the midpoint of the vector joining P(2, 3, 4) and Q(4, 1, -2).

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NCERT reference: chapter PDF pages 15, 16.

08 · Explore

Scalar (Dot) Product

The scalar product of two vectors results in a scalar value and is a measure of how much one vector goes in the direction of another.

The dot product of two vectors a⃑ and b⃑ is defined as a⃑ ⋅ b⃑ = |a⃑| |b⃑| cos θ, where θ is the angle between them. If the dot product is zero for two non-zero vectors, they are perpendicular (θ = 90°).

In component form, if a⃑ = a₁î + a₂ĵ + a₃k̂ and b⃑ = b₁î + b₂ĵ + b₃k̂, then a⃑ ⋅ b⃑ = a₁b₁ + a₂b₂ + a₃b₃. This is because î⋅î = ĵ⋅ĵ = k̂⋅k̂ = 1 and î⋅ĵ = ĵ⋅k̂ = k̂⋅î = 0.

The projection of vector a⃑ on vector b⃑ is given by (a⃑ ⋅ b⃑) / |b⃑|. This represents the scalar length of the 'shadow' of a⃑ on the line of b⃑.

Angle Between Vectors

a⃑ = î + ĵ - k̂, b⃑ = î - ĵ + k̂

1. a⃑ ⋅ b⃑ = (1)(1) + (1)(-1) + (-1)(1) = 1 - 1 - 1 = -1. 2. |a⃑| = √3, |b⃑| = √3. 3. cos θ = (a⃑ ⋅ b⃑) / (|a⃑||b⃑|) = -1 / (√3 ⋅ √3) = -1/3. θ = cos⁻¹(-1/3).

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Question

If |a⃑| = 3, |b⃑| = 4, and a⃑ ⋅ b⃑ = 6, find the angle between them.

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NCERT reference: chapter PDF pages 18, 19, 20.

09 · Explore

Vector (Cross) Product

The cross product of two vectors results in a new vector that is perpendicular to both original vectors.

The cross product is defined as a⃑ × b⃑ = |a⃑| |b⃑| sin θ n̂, where θ is the angle between them and n̂ is a unit vector perpendicular to the plane of a⃑ and b⃑, following the right-hand rule. If a⃑ × b⃑ = 0⃑, the vectors are parallel.

The cross product is non-commutative: a⃑ × b⃑ = -(b⃑ × a⃑). For unit vectors: î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ. If the order is reversed (e.g., ĵ × î), the result is negative (-k̂).

Geometrically, the magnitude |a⃑ × b⃑| represents the area of a parallelogram with adjacent sides a⃑ and b⃑. The area of a triangle with adjacent sides a⃑ and b⃑ is (1/2)|a⃑ × b⃑|.

Calculating Cross Product

a⃑ = 2î + ĵ + 3k̂, b⃑ = 3î + 5ĵ - 2k̂

Use determinant: î(1⋅-2 - 3⋅5) - ĵ(2⋅-2 - 3⋅3) + k̂(2⋅5 - 1⋅3) = î(-2-15) - ĵ(-4-9) + k̂(10-3) = -17î + 13ĵ + 7k̂.

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Question

What is the cross product of two parallel vectors?

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NCERT reference: chapter PDF pages 25, 26, 27, 28.

10 · Explore

Summary of Vector Formulas

A concise reference for key vector operations and algebraic properties.

The cross product of vectors a⃑ = a₁î + a₂ĵ + a₃k̂ and b⃑ = b₁î + b₂ĵ + b₃k̂ can be calculated using the determinant of a 3x3 matrix where the first row contains unit vectors î, ĵ, k̂, the second row contains components of a⃑, and the third row contains components of b⃑.

The position vector of a point P(x, y, z) is r = xî + yĵ + zk̂ with magnitude |r| = √(x² + y² + z²). The dot product is a⃑ ⋅ b⃑ = a₁b₁ + a₂b₂ + a₃b₃.

Determinant Formula

a⃑ × b⃑ = | î ĵ k̂ ; a₁ a₂ a₃ ; b₁ b₂ b₃ |

This determinant expansion provides the component form of the cross product: (a₂b₃ - a₃b₂)î - (a₁b₃ - a₃b₁)ĵ + (a₁b₂ - a₂b₁)k̂.

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Question

What is the relation between direction cosines l, m, n?

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NCERT reference: chapter PDF pages 29, 36, 37, 38.

Vector Algebra Class 12 Notes

Chapter mind map

Vector Fundamentals

Orientation and Components

Geometric Operations

Vector Products

Chapter notes

Basic Concepts and Definitions

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Direction Cosines and Ratios

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Types of Vectors

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Addition of Vectors

Triangle Law Sequence

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Components of a Vector

Finding a Unit Vector

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Vector Joining Two Points

Vector Calculation

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Section Formula

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Scalar (Dot) Product

Angle Between Vectors

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Vector (Cross) Product

Calculating Cross Product

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Summary of Vector Formulas

Determinant Formula

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