Resolving a Vector Parallel and Perpendicular to a Given Vector

Suppose we want to express a vectorin terms of components perpendicular and parallel to some other vector

We can peg the vectors to the origin.

If the angle between them is whereis the anticlockwise direction fromtowe can write the direction vector ofasThe component ofin the direction ofis then

To find the component ofperpendicular todraw in the perpendicular toas shown below.

The component ofin the direction perpendicular tois then

Iftake the negative square root and iftake the positive square root.

Example: Find the components ofperpendicular and parallel to