Vectors in Two Dimensions

What Is a Vector?

A vector is a quantity with both magnitude (length) and direction:

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

where $a$ is the horizontal component and $b$ is the vertical component.

Magnitude

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

Example: $|\langle 3, 4 \rangle| = \sqrt{9 + 16} = 5$

Unit Vectors

A unit vector has magnitude 1:

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

Standard unit vectors: $\hat{i} = \langle 1, 0 \rangle$ and $\hat{j} = \langle 0, 1 \rangle$

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

Vector Operations

Addition & Subtraction

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

Scalar Multiplication

$c\vec{v} = \langle cv_1, cv_2 \rangle$

Dot Product

$\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2$

Geometric Form

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

Key Properties

• $\vec{u} \cdot \vec{v} = 0 \iff$ vectors are orthogonal (perpendicular)
• $\vec{u} \cdot \vec{v} > 0 \iff$ angle is acute
• $\vec{u} \cdot \vec{v} < 0 \iff$ angle is obtuse

Angle Between Vectors

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

Vector Projection

The projection of $\vec{u}$ onto $\vec{v}$:

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

Scalar projection (component of $\vec{u}$ along $\vec{v}$):

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

Applications

Work

$W = \vec{F} \cdot \vec{d} = |\vec{F}||\vec{d}|\cos\theta$

Navigation & Physics

Decompose forces, velocities, and displacements into component form for analysis.

AP Precalculus Tip: Vectors appear in Unit 4. Master component operations, dot product, and understanding when vectors are perpendicular.