Question

A vector A has components Ax = −5.50 m and Ay = 7.50 m. Find the magnitude (in m) and the direction (in degrees counterclockwise from the +x-axis) of the vector.magnitude mdirection ° counterclockwise from the +x-axis

Answer 1

The x-component of vector A is Ax = -5.50 m

The y-component of vector A is Ay = 7.50 m

We are asked to find the magnitude and the direction of the vector (counterclockwise from the +x-axis)

The magnitude of vector A is given by

$\begin{gathered} |A|=\sqrt[]{A_x+A_y} \\ |A|=\sqrt[]{(-5.50)^2+(7.50)^2_{}} \\ |A|=\sqrt[]{30.25+56.25} \\ |A|=\sqrt[]{86.50} \\ |A|=9.3\; m \end{gathered}$

So, the magnitude of vector A is 9.3 m

First, let us draw a diagram to better understand the problem.

The direction of vector A is given by

$\begin{gathered} \tan \theta=\frac{A_y}{A_x} \\ \tan \theta=\frac{7.50}{-5.50} \\ \theta=\tan ^{-1}(\frac{7.50}{-5.50}) \\ \theta=-53.75\degree \end{gathered}$

From the diagram, notice that θ lies in the second quadrant.

So, the counterclockwise direction from the positive x-axis is given by

$180\degree-53.75\degree=126.25\degree$

So, the direction is 126.25° counterclockwise from the +x-axis.