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When are two nonzero vectors orthogonal to each other?
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Remember $$\theta=\cos ^{-1}\left(\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}\right)$$ It follows from this that $\theta=\pi / 2$ if and only if $\mathbf{u} \cdot \mathbf{v}=0$. Thus, Two nonzero vectors $\mathbf{u}$ and $\mathbf{v}$ in $R^{n}$ are said to be orthogonal (or perpendicular) if $\mathbf{u} \cdot \mathbf{v}=0 .$ We will also agree that the zero vector in $R^{n}$ is orthogonal to every vector in $R^{n}$.
by Platinum (118,804 points)

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