4.6 Vectors

What is a vector?

A vector is a quantity with both magnitude and direction, represented as an arrow. It's written in component form ⟨a, b⟩ or as ai + bj. Unlike a point, a vector describes displacement — how far and in what direction — not a fixed location in the plane.

Magnitude and direction

For vector ⟨a, b⟩, the magnitude (length) is ||v|| = √(a² + b²). The direction angle θ satisfies tan θ = b/a. You can convert between component form and magnitude/direction form using v = ⟨||v||cos θ, ||v||sin θ⟩, similar to polar-rectangular conversions.

Vector operations

Addition: add corresponding components, ⟨a,b⟩ + ⟨c,d⟩ = ⟨a+c, b+d⟩ — geometrically, place tail-to-head (tip-to-tail). Scalar multiplication: k⟨a,b⟩ = ⟨ka, kb⟩ stretches/shrinks (and flips if k<0) the vector. These operations model combined velocities, forces, and displacements in real-world contexts.

Key terms

Vector

A quantity with both magnitude and direction, often represented as an arrow or in component form ⟨a, b⟩.

Magnitude of ⟨a, b⟩

||v|| = √(a² + b²), the length of the vector.

Vector addition

Add corresponding components: ⟨a,b⟩ + ⟨c,d⟩ = ⟨a+c, b+d⟩; geometrically, tip-to-tail.

Scalar multiplication of a vector

k⟨a,b⟩ = ⟨ka, kb⟩; scales the magnitude, and reverses direction if k is negative.

Direction angle of a vector

θ where tan θ = b/a; used to convert between component form and magnitude-direction form.

Try an AP-style question

What is the magnitude of ⟨3, 4⟩?

1. 5
2. 1
3. 7
4. 25