AP® Precalculus · Unit 4 · Topic 4.8
Unit Vectors and Components
A unit vector has length one and shows pure direction with no size attached. Divide any vector by its own magnitude to shrink it to length one while keeping its direction.
Direction alone doesn't need any particular length to describe it, so mathematicians agree on length one as the standard. That way every direction gets exactly one arrow representing it, and comparing directions becomes comparing these standardized arrows instead of arrows of all different sizes. A unit vector strips away how far something moved or how strong a force is, leaving only which way it points.
To build a unit vector from any vector v, divide v by its own magnitude: u = v / |v|. This works because scaling a vector by a positive number never changes its direction, only its length — and dividing by the magnitude is exactly the scale factor that forces the new length to equal one. Check it: the magnitude of u = v/|v| is |v|/|v| = 1, no matter what v was.
Once you have a unit vector, i and j give you a compact way to write it. The vector i = (1,0) points purely along the x-axis and j = (0,1) points purely along the y-axis, so any vector (a, b) can be written as a i + b j. The coefficients a and b are just the vector's x-component and y-component restated in this notation — nothing new is happening, just a different costume for the same pair of numbers. This connects straight to magnitude and direction: given a i + b j, the magnitude is still √(a2 + b2), and dividing through by that magnitude turns it into a unit vector in the same direction.
Key terms
- Unit vector
- A vector with magnitude exactly 1, used to represent pure direction.
- Magnitude
- The length of a vector, found from its components using √(a2+b2).
- i and j
- The unit vectors (1,0) and (0,1) pointing along the x-axis and y-axis, used to write any vector as a i + b j.
- Component
- One of the coordinate values of a vector, such as the a or b in a i + b j.
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End of topic.