Plane Vectors

Master 2D vectors: concepts, operations, coordinate representations, and the powerful dot product.

Geometry

Intermediate

~60 min

Vector Calculator

Calculate magnitude, add, subtract, scale vectors, and check collinearity/perpendicularity

Vector A (v₁)

Vector B (v₂)

Enter vectors to see operations

1. What is a Vector?

A vector is a quantity with both magnitude (size) and direction.

Notation

$\vec{a}$ or a (bold) — vector a
$|\vec{a}|$ or $\|\vec{a}\|$ — magnitude of a
$\vec{AB}$ — vector from A to B

Special Vectors

Zero vector $\vec{0}$ : magnitude 0, no direction
Unit vector: magnitude = 1
Standard basis: $\hat{i} = (1,0)$ , $\hat{j} = (0,1)$

2. Vector Operations

Addition

\vec{a} + \vec{b} = (a_1+b_1, a_2+b_2)

Add component by component. Geometrically: place tail of b at head of a.

Subtraction

\vec{a} - \vec{b} = (a_1-b_1, a_2-b_2)

Subtract component by component.

Scalar Multiplication

k\vec{a} = (ka_1, ka_2)

Multiply each component by scalar k. Negative k reverses direction.

Magnitude

|\vec{a}| = \sqrt{a_1^2 + a_2^2}

Length of the vector (always non-negative).

3. Coordinate Representation

Vector from Point A to Point B

\vec{AB} = (x_B - x_A, y_B - y_A)

Position Vector

Vector from origin O to point P(x, y):

\vec{OP} = (x, y)

Unit Vector

To find the unit vector in direction of a:

\hat{a} = \frac{\vec{a}}{|\vec{a}|}

4. Dot Product

\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 = |\vec{a}||\vec{b}|\cos\theta

The dot product is a scalar (number), not a vector.

Properties

Commutative: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$
Distributive: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$
$\vec{a} \cdot \vec{a} = |\vec{a}|^2$

Key Applications

Find angle: $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$
Perpendicular test: $\vec{a} \perp \vec{b} \iff \vec{a} \cdot \vec{b} = 0$
Projection calculations

5. Applications

Physics

Force, velocity, acceleration, displacement are all vectors.

Computer Graphics

Position, movement, lighting calculations use vectors.

Navigation

Direction and distance calculations.

Engineering

Structural analysis, stress calculations.

Practice Quiz

Questions

Correct

Accuracy