Skill: Determine vector magnitude

The magnitude of a vector is its length, so it is always nonnegative.

For a two-dimensional vector $\mathbf v=\langle a,b\rangle$, use

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

This comes from the Pythagorean theorem: the vector’s components form the two legs of a right triangle.

Example

Find the magnitude of

$$\mathbf v=\langle -3,4\rangle.$$

Step 1: Identify the components.

The components are $a=-3$ and $b=4$.

Step 2: Square each component.

$$(-3)^2=9 \qquad\text{and}\qquad 4^2=16$$

Notice that the negative sign disappears when squaring. A vector pointing left or right by the same amount has the same length.

Step 3: Add the squared components.

$$9+16=25$$

Step 4: Take the square root.

$$\ \vert \mathbf v\ \vert =\sqrt{25}=5$$

Therefore, the vector’s magnitude is

$$\boxed{5}.$$

The magnitude tells us the vector’s length, not its direction. The vector $\langle -3,4\rangle$ points in a different direction from $\langle 3,4\rangle$, but both have magnitude $5$.

For a three-dimensional vector $\langle a,b,c\rangle$, include all three components:

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