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?

1 Answer
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:


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#

Now, to add the two vectors, you add the one-dimensional components

(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+(-25.53)=-21.80#


#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((V_x)^2+(V_y)^2) = sqrt((-21.80)^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 #(44.37/(-21.80))# 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.