Vectors

You’ll be able to

Add vectors component by component
Compute the magnitude of a vector
Interpret a vector as a quantity with direction and length

A vector has length and direction

A vector carries both a magnitude (length) and a direction. In component form, ⟨a, b⟩ means “a units in the x-direction and b units in the y-direction.” Two vectors are equal when their components match, regardless of where they are drawn. Vectors model quantities like velocity and force, where direction matters as much as size.

Adding vectors and measuring length

To add vectors, add their matching components: ⟨1, 2⟩ + ⟨3, −1⟩ = ⟨4, 1⟩. Geometrically this is placing them tip-to-tail. To find a vector’s magnitude — its length — apply the Pythagorean theorem to the components: |⟨a, b⟩| = √(a² + b²). For ⟨3, 4⟩ this is √(9 + 16) = √25 = 5.

Vector operations

⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩ · |⟨a, b⟩| = √(a² + b²)

Addition is component-wise; magnitude is the Pythagorean length of the components. Scalar multiplication k⟨a, b⟩ = ⟨ka, kb⟩ scales the length by |k|.

Worked example

Find the magnitude of the vector u = ⟨3, 4⟩.

1.Apply the magnitude formula |⟨a, b⟩| = √(a² + b²) with a = 3 and b = 4.
2.Square the components: 3² = 9 and 4² = 16.
3.Add and take the square root: √(9 + 16) = √25 = 5.

Answer: |u| = 5. The components 3 and 4 form the legs of a right triangle whose hypotenuse — the vector’s length — is 5.

Checkpoint

Given u = ⟨3, 4⟩, what is the magnitude |u|?

Watch out

Magnitude is not the sum of the components. For ⟨3, 4⟩ the length is √(3² + 4²) = 5, not 3 + 4 = 7. You must square, add, and take the square root.

Checkpoint

Compute the vector sum ⟨1, 2⟩ + ⟨3, −1⟩.

On the exam

To decompose a vector of magnitude r at angle θ, use components ⟨r cos θ, r sin θ⟩ — the same conversion as polar coordinates. This links force/velocity problems back to unit-circle trigonometry.

Answer the 2 checkpoints as you read.