Vector Magnitude and Direction Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Find the magnitude of v=⟨3,4⟩\mathbf{v} = \langle 3, 4 \rangle.

Solution

  1. 1
    Step 1: ∥v∥=32+42=9+16=25\|\mathbf{v}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25}.
  2. 2
    Step 2: =5= 5.
  3. 3
    Check: This is a 3-4-5 right triangle ✓

Answer

55
The magnitude (length) of a vector is found using the Pythagorean theorem: ∥v∥=v12+v22\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}. This extends naturally to higher dimensions.

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 2 medium

Find the unit vector in the direction 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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