Q 27. - Calculus

Question

In Exercises 24-27, find ${\mathrm{comp}}_{\mathbf{u}}\mathbf{v}\mathbf{ }, {\mathrm{proj}}_{\mathbf{u}}\mathbf{v}$ and the component of v orthogonal to u.

Step-by-Step Solution

Answer

The values are and the component of v orthogonal to u is .

Step 1. Given information.

The given vectors are:

Step 2. Find the component of the projection of v onto u.

Then,

$$\begin{gathered} {\mathrm{comp}}_{\mathbf{u}}\mathbf{v}=\frac{\mathit{u}\mathbf{·}\mathit{v}}{||\mathit{u}||} \\ =\frac{-28}{\sqrt{14}} \\ =\frac{-28}{\sqrt{14}}\times \frac{\sqrt{14}}{\sqrt{14}} \\ =-2\sqrt{14} \end{gathered}$$

Therefore, ${\mathrm{comp}}_{\mathbf{u}}\mathbf{v}=-2\sqrt{14}$.

Step 3. Find the vector projection of v onto u .

Therefore, the vector projection of v onto u is .

Step 4. Find the component of v orthogonal to u .

The component of v orthogonal to u is $\mathbf{v}-{\mathrm{proj}}_{\mathbf{u}}\mathbf{v}$.

Therefore, the component of v orthogonal to u is .