Orthogonal

Definition

Let $\mathbf{u},\mathbf{v}$ : Nonzero vectors in $R^n$

If $\mathbf{u}\cdot \mathbf{v}=0$

Then we say $\mathbf{u}$ an $\mathbf{v}$ are orthogonal (or perpendicular).

Remark

Notice that by definition, the zero vector in $R^n$ is orthogonal to every vector in $R^n$.

Also, recall that the angle $\theta$ between two nonzero vectors in $\mathbf{u},\mathbf{v}$ in $R^n$ is defined by

$$\cos \theta =\frac{\mathbf{u}\cdot \mathbf{v}}{||\mathbf{u}||||\mathbf{v}||}$$

¹

It follows that $\theta =\pi /2⟺\mathbf{u}\cdot \mathbf{v}=0$.

Footnotes

1. Remark ↩