Definition: A unit vector is a vector of length 1; dividing any nonzero vector by its own magnitude gives the unit vector that points the same way.
What is a unit vector?
It has magnitude exactly 1 and serves as a pure direction marker. The standard ones are i = <1, 0> and j = <0, 1>, so any vector <a, b> can be written ai + bj.
How do you normalize a vector?
Divide each component by the magnitude: the unit vector in the direction of v is v / |v|.
Worked example: w = <9, 12>. Its magnitude is the square root of (81 + 144) = 15. The unit vector is <9/15, 12/15> = <0.6, 0.8>.
Check: 0.36 + 0.64 = 1.
To build a vector of length 20 in that same direction, scale: 20 times <0.6, 0.8> = <12, 16>.
What goes wrong?
Dividing by the squared magnitude, or dividing only one component. Mathbot’s rule: divide every component by the same length. A zero vector has no unit vector because you cannot divide by 0.
Where to learn it
Related terms
Unit vectors: key points
To build a unit vector, divide a nonzero vector by its magnitude: for v = ⟨3, 4⟩, the magnitude is 5, so the unit vector is ⟨3/5, 4/5⟩. A unit vector keeps the direction of the original vector but always has length 1. The standard unit vectors i = ⟨1, 0⟩ and j = ⟨0, 1⟩ point along the axes, and every vector ⟨a, b⟩ can be written ai + bj. A unit vector at angle θ from the positive x-axis is ⟨cos θ, sin θ⟩, which links vectors to the unit circle.