Chapter 9.2, Problem 28E

To determine

The component of u along v.

Answer to Problem 28E

The component of u along v is $\frac{\text{u}\cdot \text{v}}{\left|\text{v}\right|}=5.6$ .

Explanation of Solution

Given :

  $\begin{array}{l}\text{u = 7i }\\ \text{v =}8\text{i}+6\text{j}\end{array}$

Calculation:

Definition of component of u along v :

  $\frac{\text{u}\cdot \text{v}}{\left|\text{v}\right|}$

Now,

  $\begin{array}{l}\text{u =}\langle {\text{a}}_{\text{1}}{\text{,b}}_{\text{1}}\rangle \text{and}\text{v =}\langle {\text{a}}_{\text{2}}{\text{,b}}_{\text{2}}\rangle \\ \text{u}{\text{.v = a}}_1{\text{a}}_2+{\text{b}}_{\text{1}}{\text{b}}_{\text{2}}\end{array}$

So,

  $\begin{array}{l}\left(\text{u}\cdot \text{v}\right)=\left(7\times 8+\left(0\right)\times 6\right)\\ \left(\text{u}\cdot \text{v}\right)=56\end{array}$

And,

  $\begin{array}{l}\left|\text{v}\right|=\sqrt{{\text{8}}^{\text{2}}\text{+}{6}^{\text{2}}}\\ \left|\text{v}\right|=\sqrt{64+36}\\ \left|\text{v}\right|=10\end{array}$

Now,

Component of u along v:

  $\begin{array}{l}\frac{\text{u}\cdot \text{v}}{\left|\text{v}\right|}=\frac{56}{10}\\ \frac{\text{u}\cdot \text{v}}{\left|\text{v}\right|}=5.6\end{array}$