Dot product

In linear algebra, the dot product of two vectors $\vec{v} = (v_1, \cdots, v_n), \vec{w} = (w_1, \cdots, w_n)$ is defined to be $\vec{v}\cdot\vec{w} = \sum_{i=1}^n v_i w_i$. The dot product is bilinear.

For two vectors $\vec{v}$ and $\vec{w}$, we also have $\vec{v} \cdot \vec{w} = \left \Vert \vec{v} \right \Vert \left \Vert \vec{w} \right \Vert \cos \theta$, where $\theta$ is the angle that $\vec{v}$ and $\vec{w}$ form with each other. In particular, for nonzero vectors $\vec{v}$ and $\vec{w}$, we have $\vec{v} \cdot \vec{w} = 0$ if and only if $\vec{v}$ and $\vec{w}$ are perpendicular, and $\vec{v} \cdot \vec{w} = \left \Vert \vec{v} \right \Vert \left \Vert \vec{w} \right \Vert$ if and only if $\vec{v}$ and $\vec{w}$ are parallel, from which the identity $\vec{v} \cdot \vec{v} = \left \Vert \vec{v} \right \Vert ^2$ follows.

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