Definition 2.4.1. The dot product of two vectors.
Let \(\u,\v\in\R^n.\) Then the dot product or inner product of \(\u\) with \(\v,\) is
\begin{equation*}
\dpr{\u}{\v}:=\sum_{i=1}^nu_iv_i= u_1v_1+u_2v_2+\cdots+u_nv_n.
\end{equation*}
Section 2.4 Vector Properties
Theorem 2.4.2. Properties of the dot product.
Let \(\u,\v,\w\in\R^n,\) and \(a,b\in\R.\) The dot product satisfies the following properties.
\begin{align}
\|\v\|^2\amp =\dpr{\v}{\v}\tag{2.4.1}\\
\dpr{\u}{\v}\amp =\dpr{\v}{\u}\amp \text{ (commutativity) }\tag{2.4.2}\\
a(\dpr{\u}{\v})\amp =\dpr{a\u}{\v} = \dpr{\u}{a\v}\tag{2.4.3}\\
\dpr{(a\u+b\v)}{\w}\amp =\dpr{a\u}{\w}+\dpr{b\v}{\w}\amp \text{ (linearity) }\tag{2.4.4}
\end{align}
Proof.
Theorem 2.4.3. The dot product by norms and the cosine.
For nonzero vectors \(\u,\v\in\R^n,\) we have
\begin{equation}
\dpr{\u}{\v}=\|\u\|\,\|\v\|\cos\th\text{,}\tag{2.4.5}
\end{equation}
where \(\th\in[0,\pi],\) is the angle between \(\u,\) and \(\v.\)Proof.
Applying Definition 2.1.20 and the Law of Cosines to the triangle whose sides are the directed line segments in Figure 2.4.4, we obtain
\begin{align*}
\|\u-\v\|^2=\,\,\amp \|\u\|^2+\|\v\|^2-2\|\u\|\|\v\|\cos\theta.
\end{align*}
But by Theorem 2.4.2 (with some details omitted), we have
\begin{equation}
\|\u-\v\|^2=\dpr{(\u-\v)}{(\u-\v)}\,=\,\|\u\|^2+\|\v\|^2-2\dpr{\u}{\v}.\tag{2.4.6}
\end{equation}
Theorem 2.4.3 follows immediately.
Definition 2.4.5. The angle between two nonzero vectors.
The geometric notion of the angle \(\th\in[0,\pi],\) between two nonzero vectors in \(\R^2\) or \(\R^3,\) is intuitive. Whether in \(\R^2,\,\R^3\) or \(\R^n,\) we define the angle \(\th\in[0,\pi]\) between two nonzero vectors by
\begin{equation*}
\theta=\cos^{-1}\left(\frac{\dpr{\u}{\v}}{\|\u\|\,\|\v\|}\right),
\end{equation*}
the result of solving (2.4.5) for \(\theta.\) There is no angle defined between the vector \(\vec{0}\) and any other vector since \(\vec{0}\) does not lie on a unique ray emanating from \(\mathcal{O}.\)Definition 2.4.6. Orthogonality.
Two vectors \(\u,\v\in\R^n\) are orthogonal if \(\dpr{\u}{\v}=0.\) In this case we write \(\u\perp\v.\)Definition 2.4.7. Perpendicularity.
Two nonzero vectors \(\u,\v\in\R^2\) or \(\R^3\) are perpendicular if \(\dpr{\u}{\v}=0.\) In this case we write \(\u\perp\v.\)Theorem 2.4.8. The angles between parallel, antiparallel, and nonzero orthogonal vectors.
Let \(\u,\v\in\R^n\setminus\left\{\vec{0}\right\}\) and denote the angle between \(\u\) and \(\v\) by \(\theta.\) Then\(\qquad\qquad\qquad\qquad\displaystyle{\theta=\frac{\pi}{2}}\,\,\Longleftrightarrow\,\,\u\perp\v,\)
\(\qquad\qquad\qquad\qquad\theta=0\,\,\Longleftrightarrow\,\,\u\parallel\v,\,\) and
\(\qquad\qquad\qquad\qquad\theta=\pi\,\,\Longleftrightarrow\,\,\u\upharpoonleft\!\downharpoonright\v.\)
Proof.
Theorem 2.4.9. Parallelity and antiparallelity via the dot product.
Let \(\u,\v\in\R^n\setminus\left\{\vec{0}\right\}.\) Then
\(\qquad\qquad\qquad\qquad\u\parallel\v\,\,\Longleftrightarrow\,\,\dpr{\u}{\v}=\|\u\|\,\|\v\|,\) and
\(\qquad\qquad\qquad\qquad\u\upharpoonleft\!\downharpoonright\v\,\,\Longleftrightarrow\,\,\dpr{\u}{\v}=-\|\u\|\,\|\v\|.\)
Proof.
Theorem 2.4.10. Characterizing \(\boldsymbol{\vec{0}}\) by the dot product.
Let \(\u\in\R^n.\) Then \(\u=\vec{0}\) if and only if for all \(\v\in\R^n\) we have \(\dpr{\u}{\v}=0.\)Proof.
Definition 2.4.11. Orthogonal sets.
Let \(n\in\N,\) fix \(k\le n,\) and let \(\boldsymbol{\Phi}=\{\v_1,\,\v_2,\,\ldots,\,\v_k\},\) be a set of vectors in \(\R^n.\) We say that \(\boldsymbol{\Phi}\) is an orthogonal set or more simply that \(\boldsymbol{\Phi}\,\) is orthogonal, if \(i\ne j\,\,\Rightarrow\,\,\v_i\perp\v_j.\) The vectors in an orthogonal set are said to be mutually orthogonal.Theorem 2.4.12. General Pythagorean Theorem on \(\boldsymbol{\R^n}\).
Let \(n\in\N,\) fix \(k\le n,\) and let \(\boldsymbol{\Phi}=\left\{\boldsymbol{v_1},\boldsymbol{v_2},\ldots,\boldsymbol{v_k}\right\}\) be orthogonal in \(\R^n.\) Then
\begin{equation*}
\left\|\boldsymbol{v_1}+\boldsymbol{v_2}+\cdots+\boldsymbol{v_k}\right\|^2=\left\|\boldsymbol{v_1}\right\|^2+\left\|\boldsymbol{v_2}\right\|^2+\cdots+\left\|\boldsymbol{v_k}\right\|^2.
\end{equation*}
Proof.
\begin{align*}
\left\|\sum_{i=1}^k\boldsymbol{v_i}\right\|^2=\amp \dpr{\left(\sum_{i=1}^k\boldsymbol{v_i}\right)}{\left(\sum_{j=1}^k\boldsymbol{v_j}\right)}\,=\,\sum_{i=1}^k\sum_{j=1}^k\left(\dpr{\boldsymbol{v_i}}{\boldsymbol{v_j}}\right)\\
=\amp\,\sum_{i=1}^k\left(\dpr{\boldsymbol{v_i}}{\boldsymbol{v_i}}\right)\,=\,\sum_{i=1}^k\left\|\boldsymbol{v_i}\right\|^2.
\end{align*}
Definition 2.4.13. Orthonormal vectors and orthonormal sets.
If \(\u,\,\v,\) are orthogonal unit vectors, then we say \(\u,\v\) are orthonormal vectors.A set \(\boldsymbol{\Phi}\) is called orthonormal if it contains only mutually orthogonal unit vectors.
Theorem 2.4.14. The Cauchy-Schwarz inequality.
Let \(\u,\v\in\R^n.\) Then
\begin{equation*}
\left|\dpr{\u}{\v}\right|\le\|\u\|\,\|\v\|.
\end{equation*}
Proof.
Theorem 2.4.15. The triangle inequality.
Let \(\u,\v\in\R^n.\) Then
\begin{equation*}
\|\u+\v\|\le\|\u\|+\|\v\|.
\end{equation*}
Proof.
Let \(\u,\v\in\R^n.\) We will prove
\begin{equation}
\|\u+\v\|^2\le\left(\|\u\|+\|\v\|\right)^2\tag{2.4.7}
\end{equation}
first, and then take the square root of both sides. The LHS of (2.4.7) is
\begin{align*}
\|\u+\v\|^2=\amp\,\,\sum_{i=1}^n(u_i+v_i)^2\amp \amp \text{by }\knowl{./knowl/xref/x2norm.html}{\text{Def. 2.1.20}}\\
=\amp\,\,\sum_{i=1}^n\left(u_i^2+2u_iv_i+v_i^2\right)\amp \amp \text{by basic algebra }\\
=\amp\,\,\sum_{i=1}^nu_i^2+2\sum_{i=1}^nu_iv_i+\sum_{i=1}^nv_i^2\amp \amp \text{by basic algebra }\\
=\amp\,\,\|\u\|^2+2\u\cdot\v+\|\v\|^2\amp \amp\text{by }\knowl{./knowl/xref/x2norm.html}{\text{Def. 2.1.20}}\text{ and }\knowl{./knowl/xref/dotprod.html}{\text{Def. 2.4.1}}\\
\le\amp\,\,\|\u\|^2+2\|\u\|\|\v\|+\|\v\|^2\amp \amp\text{by }\knowl{./knowl/xref/Cauchy-Schwarz.html}{\text{Thm. 2.4.14}}\\
=\amp\,\,\left(\|\u\|+\|\v\|\right)^2\amp \amp\text{by basic algebra }
\end{align*}
