Vector addition?

Vector A has a magnitude of 13 units at a direction of 250 degrees and vector B has a magnitude of 27 units at 330 degrees, both measured with respect to the positive x axis


first what is the sum of A and B


second what is the vector C=A-B


how is this done

Vector addition?
I would convert both vectors to cartesian form using simple trigonometry. 250 degrees is a 3rd quadrant angle so its sine and cosine are -cos(70) and -sin(70) respectively.





A is (-13 cos(70), -13 sin(70)) = (-4.4463, -12.2160)





A quick check here: square these numbers and add them together, and you get 169.0000, which means I've got the right answer so far, by Pythagoras.





Similarly


B is (-27 cos(60), +27 sin(60)) = (-13.5, +23.383)





Same check still says I am on the right track.


But note that these things are vectors so you write them with the x displacement above the y displacement, not on one line as I have because I'm using a dumb keyboard instead of a pencil like real mathematicians do.





A + B = (-4.4463, -12.2160) + (-13.5, +23.383)


= (-16.6623, +11.167)


= (-16.66, +11.17) to 2 dp





C = A - B = (-4.4463, -12.2160) - (-13.5, +23.383)


= (+9.0537, -35.599)


= (+9.05, -35.60) to 2 dp





If you prefer, you can convert the vectors back into polar form





A + B = (-16.6623, +11.167)


x is -ve and y is +ve so this is a 4th quadrant angle


arctan(16.6623/11.167) = arctan(0.670196) = 33.8298 deg


So the angle of A + B is 33.83 + 270 = 303.83 deg to 2 dp


The length of the vector by Pythagoras again, jolly useful fellow he must have been


16.6623^2 + 11.167^2 = 402.334 = 20.0583^2


So A + B = 20.06 in a direction 303.83 deg





Similarly


C = (+9.0537, -35.599) 2nd quadrant angle


arctan(35.599/9.0537) = 75.7308 deg


90 + 75.7308 = 165.7308


9.0537^2 + 35.599^2 = 36.7323^2


Hence the vector C is 36.73 in a direction 165.73 to 2 dp
Reply:I think that A+B=31.something








First of all draw two perpendicular lines- assume them as X and Y axes.


Now take 250 degree from +ve X axis and draw the 13 unit vector A.


Take 330 degree and braw vecor B.


Now with these two vectors as the adjascent sides make a parallelogram.


Now draw the diagonal from origin to the opposite corner of the parallelogram.


This is A+B.


The mgnitude and direction are represented by this vector if you took th first two vectors in same scale and angle.








A-B can be found out just be drawing B in the Backward direction that is at an angle of (330-180)degree


and then following above steps.





This is Geometric method.





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Now another way to find the magnitude of the resultant is





Find angle between the two vecors let it be 'Q' Q=330-250=80.


|A+B|(magnitude of A+B) = square root( (A)sqr+ (B)sqr + (2*A*B*cos(Q)) )


(A)sqr represents A*A

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