# If V= (4,-3), how do you find the direction?

Dec 3, 2016

The vector $V = \left(4 , \text{-} 3\right)$ is at an angle of -36.67° to the positive $x$-axis.

#### Explanation:

I'll presume you're looking for the angle which the vector makes with the $x$-axis.

We use our old friend $\tan$ for this.

$\tan \theta = \frac{y}{x}$
$\implies \theta = {\tan}^{-} 1 \left(\frac{y}{x}\right)$
$\textcolor{w h i t e}{\implies \theta} = {\tan}^{-} 1 \left(\frac{- 3}{4}\right) = {\tan}^{-} 1 \left(- 0.75\right)$
$\textcolor{w h i t e}{\implies \theta} = - 0.64$
color(white)(=>theta)approx-36.67°

Because ${\tan}^{-} 1$ can only return angles between -90° and +90° (that is, in ${Q}_{\text{IV}}$ or ${Q}_{\text{I}}$), we then check which quadrant the point $\left(4 , \text{-} 3\right)$ is in, to confirm whether we can keep this answer or if we need to add 180°.

$\left(4 , \text{-} 3\right)$ is in quadrant 4, and so the value given by ${\tan}^{-} 1$ is fine as it is.

The vector $V = \left(4 , \text{-} 3\right)$ is at an angle of -36.67° to the positive $x$-axis.