Dot Product
Definition
A way to multiply vectors that gives a single number — measuring how much two vectors ‘agree’ in direction.
Symbol: · Unit: depends on inputs
Technical definition
a · b = |a||b|cosθ, a scalar. It projects one vector onto another. Work is the dot product of force and displacement (W = F·d) — only the force component along the motion counts.
Example
Pushing a shopping cart at an angle: only the forward component does work — W = F·d automatically extracts it via the cosine.
Common confusion
The dot product gives a scalar, not a vector — W = F·d is a number (joules), not an arrow. Students expecting a vector answer from a dot product have confused it with the cross product.