Vectors: Introduction and Operations

By the end of this lesson, you will be able to represent vectors geometrically and algebraically, perform vector addition/subtraction and scalar multiplication, find magnitude and direction angle, and decompose a vector into components.

The Big Idea

A vector is a quantity with both magnitude (size) and direction — unlike a scalar (just size). Velocity, force, displacement, and electric fields are all vectors. Vectors are the language of physics and engineering, and they provide the bridge between trigonometry and multivariable calculus.

Representing Vectors

Geometric: an arrow from an initial point to a terminal point. The length = magnitude.

Component form: $\vec{v} = \langle a, b \rangle$ where $a$ = horizontal component, $b$ = vertical component.

Unit vectors: $\hat{i} = \langle 1, 0 \rangle$ (horizontal) and $\hat{j} = \langle 0, 1 \rangle$ (vertical).

$\vec{v} = \langle a, b \rangle = a\hat{i} + b\hat{j}$

Finding Components from Geometry

Given initial point $P(x_1, y_1)$ and terminal point $Q(x_2, y_2)$:

$$\vec{PQ} = \langle x_2 - x_1, y_2 - y_1 \rangle$$

Given magnitude $|\vec{v}|$ and direction angle $\theta$ (from positive $x$-axis):

$$\vec{v} = \langle |\vec{v}|\cos\theta, |\vec{v}|\sin\theta \rangle$$

Magnitude

$$|\vec{v}| = \sqrt{a^2 + b^2}$$

Direction Angle

$$\theta = \tan^{-1}\left(\frac{b}{a}\right)$$

Adjust for the quadrant using the component signs.

Vector Operations

Addition: $\langle a, b \rangle + \langle c, d \rangle = \langle a+c, b+d \rangle$

Subtraction: $\langle a, b \rangle - \langle c, d \rangle = \langle a-c, b-d \rangle$

Scalar multiplication: $k\langle a, b \rangle = \langle ka, kb \rangle$

Geometric addition (Parallelogram Law): place vectors tail-to-tail; the diagonal of the parallelogram is the sum.

Geometric addition (Triangle Rule): place vectors tip-to-tail; the vector from initial to final is the sum.

Unit Vectors

A unit vector has magnitude 1. To find the unit vector in the direction of $\vec{v}$: