Vectors

Vector Basics

A vector has both magnitude (length) and direction.

Notation

• Bold: $\mathbf{v}$ or $\mathbf{u}$
• Arrow: $\vec{v}$
• Component form: $\mathbf{v} = \langle a, b \rangle$ or $\mathbf{v} = a\mathbf{i} + b\mathbf{j}$

Position Vector

A vector from origin to point $(a, b)$:

$\mathbf{v} = \langle a, b \rangle$

Vector Operations

Addition

$\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle$

Subtraction

$\mathbf{u} - \mathbf{v} = \langle u_1 - v_1, u_2 - v_2 \rangle$

Scalar Multiplication

$c\mathbf{v} = \langle cu_1, cu_2 \rangle$

• $c > 0$: Same direction, scaled
• $c < 0$: Opposite direction, scaled
• $c = 0$: Zero vector

Magnitude (Length)

For $\mathbf{v} = \langle a, b \rangle$:

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

For $\mathbf{v} = \langle a, b, c \rangle$ in 3D:

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

Unit Vector

A vector with magnitude 1 in the direction of $\mathbf{v}$:

$\hat{\mathbf{u}} = \frac{\mathbf{v}}{|\mathbf{v}|} = \left\langle \frac{a}{|\mathbf{v}|}, \frac{b}{|\mathbf{v}|} \right\rangle$

Standard Unit Vectors

$\mathbf{i} = \langle 1, 0 \rangle \quad \text{(x-direction)}$

$\mathbf{j} = \langle 0, 1 \rangle \quad \text{(y-direction)}$

$\mathbf{k} = \langle 0, 0, 1 \rangle \quad \text{(z-direction, 3D)}$

Direction Angle

For $\mathbf{v} = \langle a, b \rangle$, the direction angle $\theta$:

$\tan\theta = \frac{b}{a}$

$\theta = \arctan\left(\frac{b}{a}\right)$

Component Form from Magnitude and Angle

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

$\mathbf{v} = |\mathbf{v}|\cos\theta \cdot \mathbf{i} + |\mathbf{v}|\sin\theta \cdot \mathbf{j}$

Dot Product (Scalar Product)

$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2$

Properties

$\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \quad \text{(commutative)}$

$\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} \quad \text{(distributive)}$

$c(\mathbf{u} \cdot \mathbf{v}) = (c\mathbf{u}) \cdot \mathbf{v} = \mathbf{u} \cdot (c\mathbf{v})$

$\mathbf{u} \cdot \mathbf{u} = |\mathbf{u}|^2$

Angle Between Vectors

$\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta$

$\cos\theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}$

Orthogonal Vectors

Vectors are perpendicular (orthogonal) if:

$\mathbf{u} \cdot \mathbf{v} = 0$

Cross Product (3D only)

For $\mathbf{u} = \langle u_1, u_2, u_3 \rangle$ and $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$:

$\mathbf{u} \times \mathbf{v} = \langle u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1 \rangle$

Properties

$\mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{u}) \quad \text{(anti-commutative)}$

$|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin\theta \quad \text{(magnitude)}$

$\mathbf{u} \times \mathbf{v} \perp \mathbf{u} \text{ and } \mathbf{u} \times \mathbf{v} \perp \mathbf{v} \quad \text{(perpendicular to both)}$

Projection

Projection of $\mathbf{u}$ onto $\mathbf{v}$:

$\text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2} \mathbf{v}$

Scalar projection (component):

$\text{comp}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|}$

Vector Between Two Points

Vector from point $P(x_1, y_1)$ to $Q(x_2, y_2)$:

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

Applications

• Physics: Force, velocity, acceleration
• Navigation: Displacement, heading
• Computer Graphics: Transformations, lighting
• Engineering: Stress analysis, fluid flow