Quick answer: For vector v = (a,b), magnitude ‖v‖ = √(a² + b²). It measures length and is always nonnegative. Signs affect direction but disappear after squaring for magnitude.
Worked example
Find the magnitude and a unit vector in the direction of v = (−3,4).
- Magnitude is √(9 + 16) = 5.
- Divide both components by 5.
- The unit vector is (−3/5,4/5).
Check the result
Its squared magnitude is 9/25 + 16/25 = 1, and its direction is unchanged.
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Common mistake to avoid
Do not add absolute component values: abs(−3) + abs(4) = 7 is not Euclidean length. The zero vector cannot be normalized.
Try it yourself
1. Find ‖(5,12)‖. 2. Find ‖2v‖ if ‖v‖ = 5.
Answers and reasoning
1. √169 = 13. 2. ‖2v‖ = abs(2)×5 = 10.
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