The x and y components of four vectors, $\overline{a}$,$\overrightarrow{b}$,$\overrightarrow{c}$and $\overrightarrow{d}$are given below. For which vectors will your calculator give you the correct angle$\theta$when you use it tofind $\theta$with Eq. 3-6? Answer first by examining Fig. 3-12, and then check youranswers with your calculator.

$$\begin{gathered} a_x=3a_y=3c_x=-3c_y=-3 \\ {b}_x=-3b_y=3d_x=3d_y=-3 \end{gathered}$$

$$\begin{gathered} a_x=\text{3}{a}_y=\text{3}{c}_x=-\text{3}{c}_y=-\text{3} \\ {b}_x=-\text{3}{b}_y=\text{3}{d}_x=\text{3}{d}_y=-\text{3} \end{gathered}$$

The problem involves a simple calculation of the direction of the vector. A vector's orientation, or the angle it forms with the x-axis, determines its direction. A line with an arrow on top and a fixed point at the other end is used to represent a vector. The vector's direction can be determined by looking at the direction of its arrowhead.

The angle theta is defined as the tan inverse of the ratio of the y component to the x component of the vector.

For the vector $\overrightarrow{a}$,

$\theta ={\tan }^{-1}\frac{a_y}{a_x}$

$$\begin{gathered} \theta ={\tan }^{-1}\frac{3}{3} \\ ={\tan }^{-1}\left(1\right) \\ ={45}^{\text{o}} \end{gathered}$$

As indicated by the darker portion of the graph from figure 3.12 the value of${\tan }^{-1}\left(1\right)$ which is angle 45^o fits the calculator range.

For the vector $\overrightarrow{b}$,

$\theta ={\tan }^{-1}\frac{b_y}{b_x}$

$$\begin{gathered} \theta ={\tan }^{-1}\frac{3}{-3} \\ ={\tan }^{-1}\left(-1\right) \\ =-{45}^o \end{gathered}$$

As indicated by the darker portion of the graph from figure 3.12 the value of${\tan }^{-1}\left(1\right)$which is angle - 45^o fits the calculator range.

For the vector $\overrightarrow{c}$,

$\theta ={\tan }^{-1}\frac{c_y}{c_x}$

$$\begin{gathered} \theta ={\tan }^{-1}\frac{-3}{-3} \\ ={\tan }^{-1}\left(1\right) \\ ={45}^o \end{gathered}$$

As indicated by the darker portion of the graph from figure 3.12 the value of${\tan }^{-1}\left(1\right)$which is angle 45^o fits the calculator range.

For vector 4,

$\theta ={\tan }^{-1}\frac{d_y}{d_x}$

$$\begin{gathered} \theta ={\tan }^{-1}\frac{-3}{3} \\ ={\tan }^{-1}\left(-1\right) \\ =-{45}^o \end{gathered}$$

As indicated by the darker portion of the graph from figure 3.12 the value of${\tan }^{-1}\left(1\right)$which is angle - 45^o fits the calculator range.

Thus, from the figure, it can be said that for all vectors the calculator give you the correct angle when you use it to find$\theta$