Chapter 8.2, Problem 48E

To determine

To prove: the given condition.

Answer to Problem 48E

See explanation

Explanation of Solution

Given information:

Any vector $\overrightarrow{\text{v}}$ can be written as $\langle \left|\overrightarrow{\text{v}}\right|\cos \theta ,\left|\overrightarrow{\text{v}}\right|sin\theta \rangle$ .

Calculation:

Let the tail of vector be $\left(x_1,y_1\right)$ and tip at $\left(x_2,y_2\right)$ . Sketch of the vector will be as follows:

  

The vector can be written as follows:

  $\overrightarrow{\text{v}}=\langle x_2-x_1,y_2-y_1\rangle$

The value $\left(x_2-x_1\right)\text{ and }\left(y_2-y_1\right)$ of can be evaluated as follows:

  $\begin{array}{c}\cos \theta =\frac{x_2-x_1}{\left|\overrightarrow{\text{v}}\right|}\\ \left|\overrightarrow{\text{v}}\right|\cos \theta =x_2-x_1\\ \\ \sin \theta =\frac{y_2-y_1}{\left|\overrightarrow{\text{v}}\right|}\\ \left|\overrightarrow{\text{v}}\right|sin\theta =y_2-y_1\end{array}$

Put the values of the in the equation of vector $\overrightarrow{\text{v}}$ .

  $\begin{array}{c}\overrightarrow{\text{v}}=\langle x_2-x_1,y_2-y_1\rangle \\ \overrightarrow{\text{v}}=\langle \left|\overrightarrow{\text{v}}\right|\cos \theta ,\left|\overrightarrow{\text{v}}\right|sin\theta \rangle \end{array}$

Thus, it is proved that any vector $\overline{\text{v}}$ can be written as $\langle \left|\overline{\text{v}}\right|\cos \theta ,\left|\overline{\text{v}}\right|sin\theta \rangle$ .