Work

The dot product of a constant force and a displacement, $W=\mathbf{F}\cdot\mathbf{d}$ .

The work done by a constant force $\mathbf{F}$ acting along a straight displacement $\mathbf{d}$ is $W=\mathbf{F}\cdot\mathbf{d}$ . Only the force's shadow along $\mathbf{d}$ (its scalar projection onto $\mathbf{d}$ , scaled by $|\mathbf{d}|$ ) does any work — the dot product computes exactly that in one step, without ever finding the angle between force and displacement explicitly.

Examples

A force $\mathbf{F}=\langle 8,3,-1\rangle$ newtons acting along a displacement $\mathbf{d}=\langle 5,0,2\rangle$ meters does $W=\mathbf{F}\cdot\mathbf{d}=38$ joules of work.
A sail catching wind at an angle to a boat's heading, or a tow cable pulling a sled uphill at an angle to the slope, both need the dot-product version of work.