The dot product

Find $\left[\begin{array}{r}1\\ 2\\ 3\\ 4\end{array}\right]\boldsymbol{\cdot}\left[\begin{array}{r}2\\ 0\\ 1\\ 3\end{array}\right]$.

Find $\cos\theta$ where $\theta$ is the angle between the vectors

Find $\cos\theta$ where $\theta$ is the angle between the vectors

Use the formula given in Proposition 2.47 to verify the Cauchy Schwarz inequality and to show that equality occurs if and only if one of the vectors is a scalar multiple of the other.

Show that the triangle with vertices $A=(2,0,-3)$, $B=(5,-2,1)$ and $C=(7,5,3)$ is a right triangle.

Find $\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$ where $\mathbf{w}=\left[\begin{array}{r}1\\ 0\\ -2\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{r}1\\ 2\\ 3\end{array}\right]$.

Find $\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$ where $\mathbf{w}=\left[\begin{array}{r}1\\ 2\\ -2\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{r}1\\ 0\\ 3\end{array}\right]$.

Find $\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$ where $\mathbf{w}=\left[\begin{array}{r}1\\ 2\\ -2\\ 1\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{r}1\\ 2\\ 3\\ 0\end{array}\right]$.

Does it make sense to speak of $\operatorname{proj}_{\mathbf{0}}(\mathbf{w})$?

Decompose the vector $\mathbf{v}$ into $\mathbf{v}=\mathbf{a}+\mathbf{b}$ where $\mathbf{a}$ is parallel to $\mathbf{u}$ and $\mathbf{b}$ is orthogonal to $\mathbf{u}$.

Prove the Cauchy Schwarz inequality in $\mathbb{R}^{n}$ as follows. For $\mathbf{u},\mathbf{v}$ vectors, consider

Simplify using the axioms of the dot product and then put in the formula for the projection. Notice that this expression equals $0$ and you get equality in the Cauchy Schwarz inequality if and only if $\mathbf{u}=\operatorname{proj}_{\mathbf{v}}(\mathbf{u})$. What is the geometric meaning of $\mathbf{u}=\operatorname{proj}_{\mathbf{v}}(\mathbf{u})$?

Let $\mathbf{v},\mathbf{w}$ $\mathbf{u}$ be vectors. Show that $(\mathbf{w}+\mathbf{u})_{\perp}=\mathbf{w}_{\perp}+\mathbf{u}_{\perp}$, where $\mathbf{w}_{\perp}=\mathbf{w}-\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$.

Show that

and conclude every vector in $\mathbb{R}^{n}$ can be written as the sum of two vectors, one which is orthogonal and one which is parallel to the given vector.