Vector Magnitude and Direction Math Example 2

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Example 2

medium
Find the unit vector in the direction of v=⟨1,2,2⟩\mathbf{v} = \langle 1, 2, 2 \rangle.

Solution

  1. 1
    Step 1: ∥v∥=1+4+4=9=3\|\mathbf{v}\| = \sqrt{1 + 4 + 4} = \sqrt{9} = 3.
  2. 2
    Step 2: v^=v∥v∥=13⟨1,2,2⟩=⟨13,23,23⟩\hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \frac{1}{3}\langle 1, 2, 2 \rangle = \langle \frac{1}{3}, \frac{2}{3}, \frac{2}{3} \rangle.
  3. 3
    Check: ∥v^∥=1/9+4/9+4/9=9/9=1\|\hat{\mathbf{v}}\| = \sqrt{1/9 + 4/9 + 4/9} = \sqrt{9/9} = 1 ✓

Answer

⟨13,23,23⟩\langle \frac{1}{3}, \frac{2}{3}, \frac{2}{3} \rangle
A unit vector has magnitude 1 and preserves the direction of the original vector. Divide each component by the magnitude to normalize.

About Vector Magnitude and Direction

The magnitude ∥v∥\|\mathbf{v}\| is a vector's length; the direction is the angle it makes with a reference axis.

Learn more about Vector Magnitude and Direction →

More Vector Magnitude and Direction Examples

Example 1 easy

Find the magnitude of [formula].

Example 3 medium

Find the magnitude and direction angle of the vector v = <3, 4>.

Example 4 easy

Find [formula].

Example 5 hard

Find the direction angle [formula] of [formula] measured from the positive [formula]-axis.

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