Dot Product of Perpendicular Vectors

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Let $\mathbf a$ and $\mathbf b$ be vector quantities such that $\mathbf a \ne \bszero$ and $\mathbf b \ne \bszero$.

Let $\mathbf a \cdot \mathbf b$ denote the dot product of $\mathbf a$ and $\mathbf b$.


$\mathbf a \cdot \mathbf b = 0$

if and only if:

$\mathbf a$ and $\mathbf b$ are perpendicular.


By definition of dot product:

$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \cos \theta$

where $\theta$ is the angle between $\mathbf a$ and $\mathbf b$.

When $\mathbf a$ and $\mathbf b$ be perpendicular, by definition $\theta = 90 \degrees$.

The result follows by Cosine of Right Angle, which gives that $\cos 90 \degrees = 0$.