Chapter 9.2, Problem 25E

To determine

The component of u along v.

Answer to Problem 25E

The component of u along v is $\frac{\text{u}\cdot \text{v}}{\left|\text{v}\right|}=\frac{-12}{5}$ .

Explanation of Solution

Given :

  $\begin{array}{l}\text{u = }\langle 4,6\rangle \\ \text{v =}\langle 3,-4\rangle \end{array}$

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,

  $\begin{array}{l}\left(\text{u}\cdot \text{v}\right)=\left(4\times 3+6\times \left(-4\right)\right)\\ \left(\text{u}\cdot \text{v}\right)=12+\left(-24\right)\\ \left(\text{u}\cdot \text{v}\right)=-12\end{array}$

And,

  $\begin{array}{l}\left|\text{v}\right|=\sqrt{{\text{3}}^{\text{2}}\text{+}{\left(-4\right)}^{\text{2}}}\\ \left|\text{v}\right|=\sqrt{9+14}=\sqrt{25}\\ \left|\text{v}\right|=5\end{array}$

Now,

Component of u along v:

  $\frac{\text{u}\cdot \text{v}}{\left|\text{v}\right|}=\frac{-12}{5}$