Question

Vector A has a magnitude of 3.0 units and makes an angle of −90.0° with the positive x-axis. Vector B has a magnitude of 4.0 units and makes an angle of −120° with the positive x-axis. What is the direction of the vector sum A + B relative to the positive x-axis .

Answer 1

Given data:

* The magnitude of the vector A is 3 units.

* The magnitude of the vector B is 4 units.

* The direction of vector A with the positive x-axis is - 90 degree.

* The direction of the vector B with the positive x-axis is - 120 degree.

Solution:

The diagrammatic representation of the given vectors is,

The resultant horizontal components of both the vectors is,

$\begin{gathered} X=3\cos (90^{\circ})+4\cos (120^{\circ}) \\ X=0+4\cos (90^{\circ}+30^{\circ}) \\ X=-4\sin (30^{\circ}) \\ X=-2\text{ units} \end{gathered}$

The resultant vertical component of both the vectors is,

$\begin{gathered} Y=-3-4\cos (120^{\circ}-90^{\circ}) \\ Y=-3-4\cos (30^{\circ}) \\ Y=-6.46\text{ units} \end{gathered}$

The direction of the resultant of both the vectors is,

$\tan (\theta)=\frac{Y}{X}$

Substituting the known values,

$\begin{gathered} \tan (\theta)=\frac{-6.46}{-2} \\ \tan (\theta)=3.23 \\ \theta=-107.2^{\circ} \end{gathered}$

Here negative sign indicates that resultant vector is present in third quadrant and the angle of resultant from the positive x-axis is measured in anticlockwise direction.

Thus, the direction of sum A+B (or resultant of vector A and B) is -107.2 degree or approx -107 degree.

Hence, option A is the correct answer.