Precalculus · Chapter 6

Vectors

Magnitude, addition, and the dot product.

The idea

A vector ⟨x, y⟩ represents a quantity with both size and direction — like a displacement or a force. Its magnitude (length) comes straight from the Pythagorean theorem:

||v|| = √(x² + y²)

Vectors add component by component:

⟨x₁, y₁⟩ + ⟨x₂, y₂⟩ = ⟨x₁ + x₂, y₁ + y₂⟩

The dot product of two vectors combines them into a single number — useful for finding angles between vectors and checking perpendicularity:

⟨a₁, a₂⟩ · ⟨b₁, b₂⟩ = a₁b₁ + a₂b₂

Practice problems

Problem 1

Given the vector v = ⟨3, 4⟩, find its magnitude.

Show answer

||v|| = √(3² + 4²) = √(9 + 16) = √25 = 5.

Problem 2

Given u = ⟨2, −1⟩ and v = ⟨−3, 5⟩, find u + v.

Show answer

u + v = ⟨2 + (−3), −1 + 5⟩ = ⟨−1, 4⟩.

Problem 3

Find the dot product of a = ⟨4, 1⟩ and b = ⟨2, 3⟩.

Show answer

a · b = (4)(2) + (1)(3) = 8 + 3 = 11.

Problem 4

A vector has magnitude 10 and points 30° above the horizontal. Find its horizontal and vertical components, to the nearest tenth.

Show answer

x = 10 · cos(30°) ≈ 8.7, y = 10 · sin(30°) = 5.0.