Inner product and norm
We introduce the standard inner product and compute the distance and the angle between to vectors.
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Study Inner product and norm
Inner product, norm and angle
Definition 1. Consider any vectors \[\mathbf{v}=\begin{pmatrix} v_1\\v_2\\ \vdots \\ v_n\end{pmatrix} \ \text{ and } \ \mathbf{w}=\begin{pmatrix} w_1\\w_2\\ \vdots \\ w_n\end{pmatrix}\] in \(\mathbb{R}^n\). Then the (standard) inner product of \(\mathbf{v}\) and \(\mathbf{w}\) is defined as follows: \[\langle \mathbf{v},\mathbf{w} \rangle := v_1w_1 + v_2w_2 + \ldots + v_nw_n = \sum_{k=1}^n v_kw_k\ .\]
In particular, the length \(\|\mathbf{v}\|\) of \(\mathbf{v}\) (also called norm) can be written as follows: \[\|\mathbf{v}\| = \sqrt{\langle \mathbf{v},\mathbf{v} \rangle}\, .\]
Definition 2. Consider any \(\mathbf{v},\mathbf{w}\in\mathbb{R}^n\), let \(\langle \cdot,\cdot \rangle\) be the standard inner product and let \(\|\cdot\|\) denote the norm. Then
\(\|\mathbf{v}\|\) is the length of the vector \(\mathbf{v}\),
\(\|\mathbf{v}-\mathbf{w}\|\) is the distance of \(\mathbf{v}\) and \(\mathbf{w}\),
the angle \(\alpha\) between \(\mathbf{v}\) and \(\mathbf{w}\) is given by \(\cos(\alpha) = \frac{\langle \mathbf{v},\mathbf{w}\rangle}{\|\mathbf{v}\|\cdot \|\mathbf{w}\|}\) .
Example 3. Consider \(\mathbf{v}:=\begin{pmatrix} 2\\1\end{pmatrix}\) and \(\mathbf{w}:=\begin{pmatrix} 1\\3\end{pmatrix}\).
The lengths of these vectors are \[\|\mathbf{v}\| = \left\|\begin{pmatrix} 2\\1\end{pmatrix} \right\|= \sqrt{2^2 + 1^2} = \sqrt{5}\] and \[\|\mathbf{w}\| = \left\|\begin{pmatrix} 1\\3\end{pmatrix} \right\|= \sqrt{1^2 + 3^2} = \sqrt{10.}\]
Since \[\langle \mathbf{v},\mathbf{w}\rangle = \langle \begin{pmatrix} 2\\1\end{pmatrix},\begin{pmatrix} 1\\3\end{pmatrix}\rangle = 2\cdot 1 + 1\cdot 3 = 5,\] the angle between \(\mathbf{v}\) and \(\mathbf{w}\) is given by \[\cos(\alpha) = \frac{5}{\sqrt{5}\cdot \sqrt{10}} = \frac{5}{\sqrt{50}} = \sqrt{\frac{25}{50}} = \frac{1}{\sqrt{2}}.\] And hence \(\alpha = \frac{\pi}{4}\).
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