# TBIL Activities for Understanding Linear Algebra

## Section5.1The dot product

### Definition5.1.0.1.

For two-dimensional vectors $$\vvec$$ and $$\wvec\text{,}$$ their dot product $$\vvec\cdot\wvec$$ is the scalar defined to be
\begin{equation*} \vvec\cdot\wvec = \twovec{v_1}{v_2}\cdot\twovec{w_1}{w_2} = v_1w_1 + v_2w_2\text{.} \end{equation*}

### Definition5.1.0.2.

For $$n$$-dimensional vectors $$\vvec$$ and $$\wvec\text{,}$$ their dot product $$\vvec\cdot\wvec$$ is the scalar defined to be
\begin{equation*} \vvec\cdot\wvec = v_1w_1 + v_2w_2+\cdots + v_nw_n\text{.} \end{equation*}

### Activity5.1.0.1.

Suppose that $$\vvec=\twovec{3}{4}$$ and $$\wvec=\twovec{2}{-2}\text{.}$$

#### (a)

Compute the dot product $$\vvec\cdot\wvec \text{.}$$

#### (b)

Sketch the vector $$\vvec\text{.}$$ Then use the Pythagorean theorem to find the length of $$\vvec\text{.}$$

#### (c)

Compute the dot product $$\vvec\cdot\vvec\text{.}$$ How is the dot product related to the length of $$\vvec\text{?}$$

### Definition5.1.0.3.

The magnitude, or length, of $$\vvec\text{,}$$ which we denote as $$\len{\vvec}\text{,}$$ is $$\len{\vvec} = \sqrt{(v_1)^2 + (v_2)^2+\cdots + (v_n)^2}\text{.}$$
We saw in the first activity that $$\vvec\cdot\vvec = \len{\vvec}^2\text{.}$$

### Definition5.1.0.4.

Two vectors $$\vvec$$ and $$\wvec$$ are perpendicular or orthogonal if $$\vvec\cdot\wvec=0\text{.}$$

### Activity5.1.0.2.

Let $$\vvec=\twovec32$$ and $$\wvec=\twovec{-1}3\text{.}$$

#### (a)

Find the lengths $$\len{\vvec}$$ and $$\len{\wvec}$$ using the dot product.

#### (b)

Find the dot product $$\vvec\cdot\wvec\text{.}$$ Are $$\vvec$$ and $$\wvec$$ orthogonal?

#### (c)

Consider the vector $$\xvec = \twovec{-2}{3}\text{.}$$ Find $$\vvec\cdot \xvec\text{.}$$ Are $$\vvec$$ and $$\xvec$$ orthogonal?

#### (d)

For what value of $$k$$ is the vector $$\twovec6k$$ perpendicular to $$\wvec\text{?}$$

### Activity5.1.0.3.

Suppose that
\begin{equation*} \vvec=\fourvec203{-2}, \hspace{24pt} \wvec=\fourvec1{-3}41\text{.} \end{equation*}

#### (a)

Find $$\len{\vvec}\text{.}$$

#### (b)

Find $$\len{\wvec}\text{.}$$

#### (c)

Find $$\vvec\cdot\wvec\text{.}$$