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arrow_back What is a dot (inner) product?

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What is a dot (inner) product?

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Consider arbitrary vectors $u$ and $v$ in $\mathbf{R}^{n}$; say,
$$
u=\left(a_{1}, a_{2}, \ldots, a_{n}\right) \quad \text { and } \quad v=\left(b_{1}, b_{2}, \ldots, b_{n}\right)
$$
The dot product or inner product or scalar product of $u$ and $v$ is denoted and defined by
$$
u \cdot v=a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n}
$$
That is, $u \cdot v$ is obtained by multiplying corresponding components and adding the resulting products. The vectors $u$ and $v$ are said to be orthogonal (or perpendicular) if their dot product is zero- that is, if $u \cdot v=0 .$
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