📐 Vectors

1. Scalar vs Vector Quantities

In physics, we deal with two types of physical quantities based on how they are described:

Definition

Scalar Quantity: A physical quantity that has only magnitude (size or numerical value) but no direction. Examples: Mass, Time, Temperature, Speed, Energy, Distance, Work.

Vector Quantity: A physical quantity that has both magnitude and direction. Examples: Displacement, Velocity, Acceleration, Force, Momentum.

Property	Scalar Quantity	Vector Quantity
Definition	Has only magnitude	Has both magnitude & direction
Representation	Simple number with unit	Arrow with length & direction
Examples	Mass, Time, Speed, Energy	Displacement, Velocity, Force
Addition Rule	Simple arithmetic addition	Vector addition laws required
Symbol	Normal letters ($m, t, E$)	Letters with arrow ($\vec{A}$) or bold ($\mathbf{A}$)

Remember

🎯 Speed is scalar, but Velocity is vector! Distance is scalar, but Displacement is vector!

2. Representation of Vectors

Vectors can be represented in two ways:

Symbolic Representation

Arrow notation: $\vec{A}$ (arrow over the letter)
Bold letters: $\mathbf{A}$ or $\mathbf{B}$ (used in textbooks)
Magnitude: $|\vec{A}|$ or simply $A$ (without arrow)
Unit vector: $\hat{A}$ (A-hat) - indicates direction only

Graphical Representation

Vector Representation

A vector is represented by an arrow (directed line segment):

Length of arrow → Represents the magnitude
Direction of arrow → Represents the direction
Arrowhead (Head) → Shows the sense of direction
Starting point (Tail) → Point of application

3. Vector Addition

Unlike scalars, vectors cannot be added using simple arithmetic. We use special laws for vector addition.

Triangle Law of Vector Addition

Law

Statement: If two vectors are represented by two sides of a triangle taken in the same order (head to tail), then their resultant is represented by the third side of the triangle taken in the opposite order.

Steps to apply Triangle Law:

Draw the first vector $\vec{A}$
From the head of $\vec{A}$, draw the second vector $\vec{B}$
Join the tail of $\vec{A}$ to the head of $\vec{B}$
This joining line represents the resultant $\vec{R} = \vec{A} + \vec{B}$

Parallelogram Law of Vector Addition

Law

Statement: If two vectors acting at a point are represented by two adjacent sides of a parallelogram, then their resultant is represented by the diagonal of the parallelogram passing through that point.

Magnitude of Resultant

$|\vec{R}| = \sqrt{A^2 + B^2 + 2AB \cos \theta}$

Direction of Resultant (angle $\alpha$ with $\vec{A}$)

$\tan \alpha = \frac{B \sin \theta}{A + B \cos \theta}$

Where $\theta$ is the angle between vectors $\vec{A}$ and $\vec{B}$.

Special Cases of Vector Addition

Important Cases

1. Same Direction ($\theta = 0^\circ$): $R = A + B$ (Maximum resultant)

2. Opposite Direction ($\theta = 180^\circ$): $R = |A - B|$ (Minimum resultant)

3. Perpendicular ($\theta = 90^\circ$): $R = \sqrt{A^2 + B^2}$ (Pythagorean theorem)

4. Equal Vectors ($A = B$): $R = 2A \cos(\theta/2)$ (Resultant bisects the angle)

4. Resolution of Vectors

Resolution of Vectors into Components

Resolution is the reverse process of vector addition. It is the process of splitting a single vector into two or more component vectors along given directions (usually perpendicular).

Concept

When a vector $\vec{A}$ makes an angle $\theta$ with the positive x-axis:

X-Component (Horizontal): $A_x = A \cos \theta$
Y-Component (Vertical): $A_y = A \sin \theta$

Component Form

$\vec{A} = A_x \hat{i} + A_y \hat{j} = (A \cos \theta)\hat{i} + (A \sin \theta)\hat{j}$

Magnitude from Components

$|\vec{A}| = \sqrt{A_x^2 + A_y^2}$

Direction from Components

$\tan \theta = \frac{A_y}{A_x}$

Tip

🎯 Memory trick: "cos" sounds like "close" — the component close to the angle uses cosine. The component perpendicular (far) uses sine!

5. Unit Vectors

A unit vector is a vector that has a magnitude of exactly 1 (unity). It is used to specify direction only.

Unit Vector Formula

$\hat{A} = \frac{\vec{A}}{|\vec{A}|}$

Standard Unit Vectors (Cartesian)

Standard Unit Vectors (3D Coordinate System)

$\hat{i}, \hat{j}, \hat{k}$

$\hat{i}$ (i-hat) → Unit vector along positive X-axis
$\hat{j}$ (j-hat) → Unit vector along positive Y-axis
$\hat{k}$ (k-hat) → Unit vector along positive Z-axis

Each has magnitude = 1 and they are mutually perpendicular to each other.

Vector in Component Form

$\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$

📝 Solved Example

If $\vec{A} = 3\hat{i} + 4\hat{j}$, find the magnitude and unit vector.

Solution:

Magnitude: $|\vec{A}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = \mathbf{5\text{ units}}$

Unit vector: $\hat{A} = \frac{\vec{A}}{|\vec{A}|} = \frac{3\hat{i} + 4\hat{j}}{5} = \mathbf{0.6\hat{i} + 0.8\hat{j}}$

6. Dot Product (Scalar Product)

Dot Product Visualization

The dot product of two vectors is a scalar quantity. It is called scalar product because the result is a scalar, not a vector.

Definition

The dot product of two vectors $\vec{A}$ and $\vec{B}$ is defined as the product of their magnitudes and the cosine of the angle between them.

Dot Product Formula

$\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta = AB \cos \theta$

Component Form

$\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$

Finding Angle Between Two Vectors

Angle Formula

$\cos \theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|}$

Properties of Dot Product

Commutative: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$
Distributive: $\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}$
Scalar multiplication: $(m\vec{A}) \cdot \vec{B} = m(\vec{A} \cdot \vec{B})$
Self dot product: $\vec{A} \cdot \vec{A} = |\vec{A}|^2$

Unit Vector Dot Products

Remember

Same unit vectors: $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = \mathbf{1}$

Different unit vectors: $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = \mathbf{0}$

Key Condition

⚠️ Condition for Perpendicular Vectors:

If $\vec{A} \cdot \vec{B} = 0$, then $\vec{A} \perp \vec{B}$ (perpendicular)

(Because $\theta = 90^\circ$, so $\cos 90^\circ = 0$)

📝 Solved Example

Check if $\vec{A} = 2\hat{i} + 3\hat{j}$ and $\vec{B} = 3\hat{i} - 2\hat{j}$ are perpendicular.

Solution:

$\vec{A} \cdot \vec{B} = (2)(3) + (3)(-2) = 6 - 6 = \mathbf{0}$

Since $\vec{A} \cdot \vec{B} = 0$, the vectors are perpendicular! ✓

7. Cross Product (Vector Product)

Cross Product - Area of Parallelogram

The cross product of two vectors is a vector quantity. It is called vector product because the result is a vector, not a scalar.

Definition

The cross product of two vectors $\vec{A}$ and $\vec{B}$ is a vector perpendicular to both, with magnitude equal to the product of their magnitudes and the sine of the angle between them.

Cross Product Formula

$\vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin \theta \cdot \hat{n}$

Where $\hat{n}$ is the unit vector perpendicular to both $\vec{A}$ and $\vec{B}$ (found using right-hand rule).

Magnitude of Cross Product

$|\vec{A} \times \vec{B}| = AB \sin \theta$

This magnitude equals the area of the parallelogram formed by the two vectors.

Right-Hand Rule (Direction)

Right-Hand Rule for Cross Product

Rule

To find the direction of $\vec{A} \times \vec{B}$:

Point your fingers along vector $\vec{A}$
Curl your fingers toward vector $\vec{B}$ (through the smaller angle)
Your thumb points in the direction of $ \vec{A} \times \vec{B}$

Component Form (Determinant Method)

Cross Product using Determinant

$\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$

Unit Vector Cross Products

Cyclic Order of Unit Vectors (î, ĵ, k̂)

Remember

Same unit vectors: $\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = \mathbf{0}$

Cyclic order (clockwise):

$\hat{i} \times \hat{j} = \hat{k} \quad | \quad \hat{j} \times \hat{k} = \hat{i} \quad | \quad \hat{k} \times \hat{i} = \hat{j}$

Anti-cyclic order (anti-clockwise):

$\hat{j} \times \hat{i} = -\hat{k} \quad | \quad \hat{k} \times \hat{j} = -\hat{i} \quad | \quad \hat{i} \times \hat{k} = -\hat{j}$

Properties of Cross Product

Anti-commutative: $\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$ ⚠️ Order matters!
Distributive: $\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$
Scalar multiplication: $(m\vec{A}) \times \vec{B} = m(\vec{A} \times \vec{B})$
Self cross product: $\vec{A} \times \vec{A} = \mathbf{0}$

Key Condition

⚠️ Condition for Parallel Vectors:

If $\vec{A} \times \vec{B} = \mathbf{0}$, then $\vec{A} \parallel \vec{B}$ (parallel)

(Because $\theta = 0^\circ$ or $180^\circ$, so $\sin \theta = 0$)

📝 Solved Example

Check if $\vec{A} = 2\hat{i} + 4\hat{j}$ and $\vec{B} = \hat{i} + 2\hat{j}$ are parallel.

Solution:

$\vec{A} \times \vec{B} = (2\hat{i} + 4\hat{j}) \times (\hat{i} + 2\hat{j}) = (2)(2) - (4)(1) = \mathbf{0}$

Since $\vec{A} \times \vec{B} = \mathbf{0}$, the vectors are parallel! ✓

(Note: $\vec{A} = 2\vec{B}$, confirming they are parallel)

8. Dot Product vs Cross Product

Property	Dot Product ($\vec{A} \cdot \vec{B}$)	Cross Product ($\vec{A} \times \vec{B}$)
Result Type	Scalar (number)	Vector
Formula	$AB \cos \theta$	$AB \sin \theta \cdot \hat{n}$
Commutativity	Commutative ($\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$)	Anti-commutative ($\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$)
When $\theta = 0^\circ$	Maximum ($AB$)	Zero ($\mathbf{0}$)
When $\theta = 90^\circ$	Zero ($0$)	Maximum ($AB$)
Physical Examples	Work, Power	Torque, Angular Momentum
Zero Condition	Vectors are Perpendicular ($\perp$)	Vectors are Parallel ($\parallel$)

9. Quick Formula Summary

Formulas

Vector Addition (Parallelogram Law):

$|\vec{R}| = \sqrt{A^2 + B^2 + 2AB \cos \theta} \quad | \quad \tan \alpha = \frac{B \sin \theta}{A + B \cos \theta}$

Resolution of Vectors:

$A_x = A \cos \theta \quad | \quad A_y = A \sin \theta \quad | \quad A = \sqrt{A_x^2 + A_y^2}$

Unit Vector:

$\hat{A} = \frac{\vec{A}}{|\vec{A}|}$

Dot Product:

$\vec{A} \cdot \vec{B} = AB \cos \theta = A_x B_x + A_y B_y + A_z B_z \quad | \quad \text{If } \vec{A} \cdot \vec{B} = 0 \to \vec{A} \perp \vec{B}$

Cross Product:

$|\vec{A} \times \vec{B}| = AB \sin \theta \quad | \quad \text{If } \vec{A} \times \vec{B} = \mathbf{0} \to \vec{A} \parallel \vec{B}$

Pro Tips

🎯 cos is for DOT (same direction component)

🎯 sin is for CROSS (perpendicular component)

🎯 Dot product = 0 → Perpendicular

🎯 Cross product = $\mathbf{0}$ → Parallel