$\mathbf{A}·\mathbf{B}\=\left[\begin{array}{c}3\\ 7\end{array}\right]·\left[\begin{array}{c}-4\\ 5\end{array}\right]$ = $3\left(-4\right)\+7\left(5\right)\=-12\+35\=23$ 

${∥\mathbf{A}∥}^2\=\mathbf{A}·\mathbf{A}\=\left[\begin{array}{c}3\\ 7\end{array}\right]·\left[\begin{array}{c}3\\ 7\end{array}\right]\=9\+49\=58$ $\Rightarrow ∥\mathbf{A}∥\=\sqrt{58}$ 

${∥\mathbf{B}∥}^2\=\mathbf{B}·\mathbf{B}\=\left[\begin{array}{c}-4\\ 5\end{array}\right]·\left[\begin{array}{c}-4\\ 5\end{array}\right]\=16\+25\=41$ $\Rightarrow ∥\mathbf{B}∥\=\sqrt{41}$ 

$\mathrm{\theta }\={\cos }^{-1}\left(\frac{\mathbf{A}·\mathbf{B}}{∥\mathbf{A}∥ ∥\mathbf{B}∥}\right)\={\cos }^{-1}\(\frac{23}{\sqrt{58} \sqrt{41}}\)\doteq 1.0796$ radians

In Part (b), the norm of A was found to be $\sqrt{58}$ by computing $\sqrt{\mathbf{A}·\mathbf{A}}$.

In Section 1.1, the norm was defined as the square root of the sum of the squares of the components. Hence, calculate

$∥\mathbf{A}∥\=\sqrt{3^2\+7^2}\=\sqrt{9\+49}\=\sqrt{58}$ 

The vector ${\mathbf{B}}_{\mathbf{A}}$, the vector projection of B on A, is shown in Figure 1.3.1(a). It is obtained by the following calculation.

${\mathbf{B}}_{\mathbf{A}}\=\frac{\mathbf{B}·\mathbf{A}}{\mathbf{A}·\mathbf{A}}\mathbf{A}$ = $\frac{23}{58} \mathbf{A}$ = $\frac{23}{58}\left[\begin{array}{c}3\\ 7\end{array}\right]$ 

The component of B orthogonal to A is the vector ${\mathbf{B}}_{\perp \mathbf{A}}$ , shown in blue in Figure 1.3.1(a). It is obtained by the following calculation.

 ${\mathbf{B}}_{\perp \mathbf{A}}\=\mathbf{B}-{\mathbf{B}}_{\mathbf{A}}$ = $\left[\begin{array}{c}-4\\ 5\end{array}\right] - \frac{23}{58}\left[\begin{array}{c}3\\ 7\end{array}\right]$ = $\frac{1}{58}\left[\begin{array}{c}-301\\ 129\end{array}\right]$ 

Although Figure 1.3.1(a) could be constructed interactively, its construction is tedious. Below, in the section where a coded solution is given, Maple commands for drawing the figure more efficiently are given.

Begin by installing the Student MultivariateCalculus package since it contains an Angle command, which will then become available in the Context Panel.

With the Student MultivariateCalculus package installed, the norm of A can be accessed via the Context Panel or via the notation $∥\mathbf{A}∥$, as per the note on norms in Section 1.1.

There is no Context Panel option for obtaining the square of the norm. Hence, the alternate calculation shows $∥\mathbf{A}∥\=\sqrt{\mathbf{A}·\mathbf{A}}$ instead of ${∥\mathbf{A}∥}^2\=\mathbf{A}·\mathbf{A}$.

The  task template, implemented in Table 1.3.1(a), provides a solution for both ${\mathbf{B}}_{\mathbf{A}}$ and ${\mathbf{B}}_{\perp \mathbf{A}}$, and in addition, draws a graph of B and its two components.

Table 1.3.1(b) implements the calculation of ${\mathbf{B}}_{\mathbf{A}}$ via the recipe listed in Table 1.3.1.

Table 1.3.1(c) implements the calculation of ${\mathbf{B}}_{\mathbf{A}}$ via the Projection option in the Context Panel, an option available only when the Student MultivariateCalculus package is installed.

Compare this result to that provided in Table 1.3.1(a).

As per Table 1.3.2, the Student MultivariateCalculus package contains a DotProduct command defined over the real numbers.

For completeness and clarity, the Coded solutions begin with the execution of a restart command.

The following code will draw the vectors in Figure 1.3.1(a). The additional text was added to the figure  interactively via the Drawing toolbar.

Recall that the Student MultivariateCalculus package was installed in Part (a).

Recall that the Student MultivariateCalculus package was installed in Part (a).

Recall that the Student MultivariateCalculus package was installed in Part (a).

Recall that the Student MultivariateCalculus package was installed in Part (a).

Recall that the Student MultivariateCalculus package was installed in Part (a).