Determine vector magnitude

A vector’s magnitude is its nonnegative Euclidean length: for $\mathbf v=\langle a,b\rangle$ , $\|\mathbf v\|=\sqrt{a^2+b^2}$ , and for $\langle a,b,c\rangle$ , $\|\mathbf v\|=\sqrt{a^2+b^2+c^2}$ , as justified by the Pythagorean theorem. The calculation shows why component signs do not affect length, distinguishes magnitude from direction, and supports distance, displacement, and applications involving vectors; abstract norms and more advanced vector spaces are not included.

Detailed Explanation: 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}.

Learn by doing: Determine vector magnitude

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Vectors - Magnitude - Vector Notation to Magnitude (2D)

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