# Given Vector A: 25.2cm at 81.6 degrees and Vector B: 32.1 cm at 142.7 degrees. How do you add and subtract these two vectors?

Jan 22, 2017

You will need to convert both vectors into rectangular components ($x$ and $y$) so that these components can be added. Details below:

#### Explanation:

This is a problem in two-dimensional vector addition.

First, you will resolve each vector (given here in standard form) into rectangular components ($x$ and $y$).

Then, you will add together the $x -$components and add together the $y -$components. This will given you the answer you seek, but in rectangular form.

Finally, convert the resultant into standard form.

Here's how:

Resolve into rectangular components

A_x = 25.2 cos 81.6° = 25.2 (0.146) = 3.73

A_y = 25.2 sin 81.6° = 25.2 (0.989) = 24.92

B_x = 32.1 cos 142.7° = 32.1 (-0.795) = -25.53

B_y = 32.1 sin 142.7° = 32.1 (0.606) = 19.45

(If you want to subtract the second vector from the first, you would subtract the components at this stage instead.)

${V}_{x} = {A}_{x} + {B}_{x} = 3.73 + \left(- 25.53\right) = - 21.80$

and

${V}_{y} = {A}_{y} + {B}_{y} = 24.92 + 19.45 = 44.37$

This is the resultant vector in rectangular form. With a negative $x$-component and a positive $y$-component, this vector points into the 2nd quadrant. Remember this for later!

Now, convert to standard form:

$V = \sqrt{{\left({V}_{x}\right)}^{2} + {\left({V}_{y}\right)}^{2}} = \sqrt{{\left(- 21.80\right)}^{2} + {44.37}^{2}} = 49.44$

theta=tan^(-1)(44.37/(-21.80)) = -63.8°

This angle looks a bit strange! Remember, the vector was stated to point into the second quadrant. Our calculator has lost track of this when we used the ${\tan}^{- 1}$ function. It noted that the argument $\left(\frac{44.37}{- 21.80}\right)$ has a negative value, but gave us the angle of the portion of a line with that slope that would point into quadrant 4. We need to be careful not to put too much faith in our calculator in a case like this. We want the portion of the line that points into quadrant 2.

To find this angle, add 180° to the (incorrect) result above. The angle we want is 116.1°.

If you get into the habit of always drawing a reasonably accurate diagram to go along with your vector addition, you will always catch this problem when it occurs.