Chapter 11, Problem 8RE

To determine

To determine: The component form of the vector.

Answer to Problem 8RE

The required component form is $3\widehat{i}+4\widehat{j}$ .

Explanation of Solution

Given information:

The length of the vector with the direction opposite to the direction $\langle \frac{3}{5},\frac{4}{5}\rangle$ = 5 units

Formula used:

Magnitude of a vector:

  $\left|\overrightarrow{v}\right|=\sqrt{{\overrightarrow{v}}_1{}^2+{\overrightarrow{v}}_2{}^2+{\overrightarrow{v}}_3{}^2}$

Unit vector formula:

  $\overrightarrow{u}=\frac{\overrightarrow{v}}{\left|\overrightarrow{v}\right|}$

Calculation:

Consider the vector be $\overrightarrow{v}=\left(\frac{3}{5}\right)\widehat{i}+\left(\frac{4}{5}\right)\widehat{j}$ .

To find the component form of the above vector, first find the magnitude and unit vector of the vector.

Find the magnitude of the above vector using the formula $\left|\overrightarrow{v}\right|=\sqrt{{\overrightarrow{v}}_1{}^2+{\overrightarrow{v}}_2{}^2+{\overrightarrow{v}}_3{}^2}$ .

  $\begin{array}{c}\left|\overrightarrow{v}\right|=\sqrt{{\left(\frac{3}{5}\right)}^2+{\left(\frac{4}{5}\right)}^2+{\left(0\right)}^2}\\ =\sqrt{\frac{9}{25}+\frac{16}{25}}\\ =\sqrt{\frac{25}{25}}\\ =1\end{array}$

Find the unit vector using the formula $\overrightarrow{u}=\frac{\overrightarrow{v}}{\left|\overrightarrow{v}\right|}$ .

  $\begin{array}{l}\overrightarrow{u}=\frac{\left(\frac{3}{5},\frac{4}{5},0\right)}{1}\\ \overrightarrow{u}=\left(\frac{3}{5},\frac{4}{5},0\right)\end{array}$

The given length of the vector is 5 units.

Multiply 5 with the above unit vector.

  $\begin{array}{c}5\overrightarrow{u}=5\left(\frac{3}{5},\frac{4}{5},0\right)\\ =\left(\cancel{5}\left(\frac{3}{\cancel{5}}\right),\cancel{5}\left(\frac{4}{\cancel{5}}\right),5\left(0\right)\right)\\ =\left(3,4,0\right)\end{array}$

Write the component form of the vector as:

  $3\widehat{i}+4\widehat{j}+\left(0\right)\widehat{k}=3\widehat{i}+4\widehat{j}$

Hence, the required component form is $3\widehat{i}+4\widehat{j}$ .