Vectors: theory | Physics Circle

CONCEPT

Vector Basics, Unit Vectors & Components

In one line
A vector has a size and a direction and adds by the triangle law. Split it into î, ĵ, k̂ parts and every vector sum becomes simple algebra.

SCALARS AND VECTORS

2.1.1 Scalars and vectors

Scalar: magnitude only (mass, time, work, energy). Added like ordinary numbers.
Vector: magnitude and direction, and it must obey the laws of vector addition (displacement, velocity, force, momentum, torque).
Electric current has a direction but adds like a number, so it is a scalar.
Pressure, stress, moment of inertia: neither a plain scalar nor a vector (tensors).

TYPES

2.1.2 Types of vectors

Type	Meaning
Equal	same size, same direction
Parallel	same direction: A = nB, n > 0
Anti-parallel	opposite direction: A = nB, n < 0
Collinear	along the same line (parallel or anti-parallel)
Zero (null)	zero size, any direction
Coplanar	lie in one plane; any two vectors are coplanar
Polar	start from a point: displacement, force
Axial	along the axis of rotation: ω, torque, angular momentum

Parallel vectors have equal ratios of components: A_xB_x = A_yB_y = A_zB_z.
Multiplying by −1 reverses the direction.

UNIT VECTOR

2.1.3 Unit vector

Â = A|A|

A = A Â

A unit vector has size 1 and no unit: it only gives the direction.
For a unit vector a î + b ĵ + c k̂: a² + b² + c² = 1.
Unit vector along a sum: add first, then divide by the size of the sum.

ORTHOGONAL UNIT VECTORS

2.1.4 î, ĵ, k̂

Right-handed axes: curl the fingers from x to y, the thumb gives z

î, ĵ, k̂ are unit vectors along x, y and z.
They form a right-handed set.

RESOLUTION

2.1.5 Resolution into components

R split into R_x along X and R_y along Y

R_x = R cos θ, R_y = R sin θ

R = √R_x² + R_y², tan θ = R_yR_x

θ is measured from the x-axis. If the angle is given from the y-axis, the y-part takes the cosine.
A component can never be bigger than the vector itself.
Two vectors are added part by part: (A_x + B_x) î + (A_y + B_y) ĵ.

3-D COMPONENTS

2.1.6 Components in space

R is the diagonal of the box with edges R_x, R_y, R_z

R = R_x î + R_y ĵ + R_z k̂

R = √R_x² + R_y² + R_z²

DIRECTION COSINES

2.1.7 Direction cosines

cos α = R_xR, cos β = R_yR, cos γ = R_zR

cos²α + cos²β + cos²γ = 1

α, β, γ = angles with the x, y and z axes.
î + ĵ + k̂ makes the same angle with all three axes: cos α = 1√3, α ≈ 54.7°.
In a plane: cos²α + cos²β = 1, so α + β = 90°.

POSITION VECTOR

2.1.8 Position and displacement vectors

Point P(x, y, z): position vector r = x î + y ĵ + z k̂.
From (x₁, y₁, z₁) to (x₂, y₂, z₂): displacement = (x₂ − x₁) î + (y₂ − y₁) ĵ + (z₂ − z₁) k̂.

Practise PYQs on this topicVector Basics, Unit Vectors & Components · 20 questions with solutions

CONCEPT

Addition & Subtraction of Vectors

In one line
Two vectors at angle θ add to R² = A² + B² + 2AB cos θ. To subtract, add the reversed vector: cos θ just changes sign.

TRIANGLE LAW

2.2.1 Triangle law

Head of A joined to the tail of B; R closes the triangle

Place the vectors head to tail. The resultant goes from the first tail to the last head.
Vector addition is commutative and associative: A + B = B + A.

PARALLELOGRAM LAW

2.2.2 Parallelogram law

A and B from one point O; R is the diagonal OC

R = √(A² + B² + 2AB cos θ)

tan α = B sin θA + B cos θ

θ = angle between A and B (tails together). α = angle of R with A.
R leans towards the bigger vector.

SPECIAL CASES

2.2.3 Special cases

θ	R	Note
0°	A + B	maximum
90°	√A² + B²	perpendicular
120° (A = B)	A	resultant equals each vector
180°	\|A − B\|	minimum

|A − B| ≤ R ≤ A + B

Two unequal vectors can never add to zero.
Fewest vectors that add to zero: 2 if equal, 3 if unequal and coplanar, 4 if not coplanar.
If A + B = C with A² + B² = C², then A ⟂ B (e.g. 3, 4, 5).

POLYGON LAW

2.2.4 Polygon law

A, B, C, D head to tail; R closes the polygon

R = A + B + C + D

If the vectors form a closed figure taken in order, their sum is zero.
A + B + C = 0 ⇒ the three vectors are coplanar.

SUBTRACTION

2.2.5 Subtraction of vectors

A − B is A + (−B): the other diagonal

A − B = A + (−B)

|A − B| = √(A² + B² − 2AB cos θ)

tan β = B sin θA − B cos θ

Subtraction is not commutative: A − B = −(B − A).
A + B and A − B are the two diagonals of the same parallelogram.
Reversing one vector: |A + B|² + |A − B|² = 2(A² + B²).

JEE · NEET

Corner

Equal vectors at angle θ

|A + B| = 2A cos θ2

|A − B| = 2A sin θ2

Then |A − B||A + B| = tan θ2.
n equal vectors at equal angles round a point add to zero: the angle between neighbours is 2π ÷ n.

Polygon with centre O

For a regular polygon with centre O, the vectors from O to all the vertices add to zero.
So from one vertex A: sum of AX over the other vertices = nAO (n = number of vertices).

Practise PYQs on this topicAddition & Subtraction of Vectors · 31 questions with solutions

CONCEPT

Scalar (Dot) Product

In one line
A·B = AB cos θ is a number. It is zero when the vectors are perpendicular: the most asked idea in MHT CET.

DEFINITION

2.3.1 Definition

B cos θ is the part of B along A

A·B = AB cos θ

Positive for acute θ, zero for 90°, negative for obtuse θ.
Maximum AB at θ = 0°; for θ = 90° it is 0.

PROPERTIES

2.3.2 Properties

Commutative: A·B = B·A.
Distributive: A·(B + C) = A·B + A·C.
A·A = A², so A = √(A·A).
î·î = ĵ·ĵ = k̂·k̂ = 1 and î·ĵ = ĵ·k̂ = k̂·î = 0.

COMPONENT FORM

2.3.3 In components

A·B = A_xB_x + A_yB_y + A_zB_z

Multiply like components and add.
A missing component is 0.

ANGLE

2.3.4 Angle between two vectors

cos θ = A·BAB

PERPENDICULAR

2.3.5 Perpendicular vectors

A ⟂ B ⇔ A·B = 0

Use it to find an unknown component: put the dot product equal to zero.
(A + B) ⟂ (A − B) ⇔ A² − B² = 0 ⇔ |A| = |B|.
|A + B| = |A − B| ⇔ A ⟂ B.

PROJECTION

2.3.6 Projection (component) along a vector

component of A along B = A·BB = A·B̂

projection vector = (A·B̂) B̂

The component is a number; the projection vector points along B.
Zero projection ⇔ perpendicular.

IN PHYSICS

2.3.7 Dot product in physics

Quantity	As a dot product
Work	W = F·s = Fs cos θ
Power	P = F·v
Magnetic flux	φ = B·A
Dipole energy	U = −p·E, U = −M·B

Practise PYQs on this topicScalar (Dot) Product · 24 questions with solutions

CONCEPT

Vector (Cross) Product

In one line
|A × B| = AB sin θ, pointing perpendicular to both by the right-hand rule. Its size is the area of the parallelogram.

DEFINITION

2.4.1 Definition

C = A × B is perpendicular to the plane of A and B

A × B = AB sin θ n̂

n̂ is perpendicular to both A and B.
Right-hand rule: curl the fingers from A to B through the smaller angle; the thumb gives A × B.
Maximum AB at 90°; zero at 0° or 180°.

PROPERTIES

2.4.2 Properties

Not commutative: A × B = −B × A. The sizes are equal.
Distributive: A × (B + C) = A × B + A × C.
A × A = 0 (null vector).
A × B = 0 ⇔ A and B are parallel or anti-parallel.
A × B is perpendicular to A + B and to A − B.

UNIT VECTORS

2.4.3 Cross products of î, ĵ, k̂

Clockwise order gives +, the other way gives −

î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ

ĵ × î = −k̂, k̂ × ĵ = −î, î × k̂ = −ĵ

î × î = ĵ × ĵ = k̂ × k̂ = 0.

COMPONENT FORM

2.4.4 In components (determinant)

A × B = î(A_yB_z − A_zB_y) − ĵ(A_xB_z − A_zB_x) + k̂(A_xB_y − A_yB_x)

Write î ĵ k̂ in the first row, A in the second, B in the third, and expand.
Mind the minus sign on the ĵ term.

AREA

2.4.5 Area

parallelogram = |A × B|

triangle = 12|A × B|

A and B are the two adjacent sides (or the two diagonals give twice the area: area = ½|d₁ × d₂|).

DOT AND CROSS

2.4.6 Dot and cross together

tan θ = |A × B|A·B

|A × B|² + (A·B)² = A²B²

Dot = cross in size ⇒ θ = 45°. Cross = √3 × dot ⇒ θ = 60°.
Then put θ in R² = A² + B² ± 2AB cos θ.

IN PHYSICS

2.4.7 Cross product in physics

Quantity	As a cross product
Torque	τ = r × F
Angular momentum	L = r × p
Linear velocity	v = ω × r
Magnetic force	F = q(v × B)
Torque on a dipole	τ = p × E, M × B

JEE · NEET

Corner

Scalar triple product

A·(B × C) = volume of the parallelepiped

A·(B × C) = 0 ⇔ A, B, C are coplanar.
It is the determinant of the three rows of components.

Practise PYQs on this topicVector (Cross) Product · 16 questions with solutions

EXAM-DAY CARD

Just see this before the exam

Â = A|A|

unit vector

cos α = A_xA

direction cosines

R_max = A + B

addition

A·B = AB cos θ

dot product

A·B = 0

perpendicular

|A × B| = AB sin θ

cross product

FORMULA SHEET

Whole chapter on one page

Unit vector

Â = A|A|

a² + b² + c² = 1

parallel: equal component ratios

Components

A_x = A cos θ, A_y = A sin θ

A = √(A_x² + A_y² + A_z²)

cos²α + cos²β + cos²γ = 1

Addition

R² = A² + B² + 2AB cos θ

tan α = B sin θA + B cos θ

|A − B| ≤ R ≤ A + B

Subtraction

|A − B|² = A² + B² − 2AB cos θ

A = B, R = A ⇒ θ = 120°

Dot product

A·B = AB cos θ = A_xB_x + A_yB_y + A_zB_z

⟂ ⇔ A·B = 0

component = A·BB

Cross product

|A × B| = AB sin θ

î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ

area = |A × B|, triangle ½|A × B|

TOP MISTAKES

Avoid these = easy marks

MISTAKE ✗

|A − B| = √(A² + B² + 2AB cos θ)

Perpendicular ⇒ A × B = 0

ĵ × î = k̂

Unit vector of A + B = Â + B̂

CORRECT ✓

Minus sign: √(A² + B² − 2AB cos θ)

Perpendicular ⇒ A·B = 0

ĵ × î = −k̂ (anticlockwise order)

Add first, then divide by |A + B|