Two vectors are orthogonal when they meet at a right angle. For nonzero vectors, this is equivalent to

$$\mathbf u\cdot\mathbf v=0.$$

The zero vector is conventionally orthogonal to every vector because its dot product with every vector is zero.

Projection onto a direction

A vector can be separated into a component parallel to a chosen nonzero vector $\mathbf u$ and a component perpendicular to it.

The parallel component is the projection of $\mathbf v$ onto $\mathbf u$:

$$\operatorname{proj}_{\mathbf u}\mathbf v

\frac{\mathbf v\cdot\mathbf u}{\mathbf u\cdot\mathbf u}\mathbf u.$$

For a unit vector $\hat{\mathbf u}$, this simplifies to

$$\operatorname{proj}_{\hat{\mathbf u}}\mathbf v =(\mathbf v\cdot\hat{\mathbf u})\hat{\mathbf u}.$$

Parallel and perpendicular components

Once the projection is known,

$$\mathbf v_{\parallel}=\operatorname{proj}_{\mathbf u}\mathbf v,$$

and

$$\mathbf v_{\perp}=\mathbf v-\mathbf v_{\parallel}.$$

By construction,

$$\mathbf v_{\perp}\cdot\mathbf u=0.$$

This decomposition is useful whenever a vector must be resolved along and across a preferred direction, such as a force on an inclined surface or a signal relative to a chosen basis direction.