Converting between vector components and magnitude & direction review (article) | Khan Academy

Vector addition, or subtraction, is just combining steps in the various directions. then finding the direction is taking the inverse tangent of the ratio of the combined j steps over the combined i steps.

Your question says 2A – B. if this were just A – B you would do 12i – 16j -(-24i + 10i). Here A is multiplied by 2, so let’s get that done first.

2A
2(12i-16j)
24i – 32j

There, now we can do the subtraction part

2A – B
24i – 32j – (-24i + 10j)
24i – 32j + 24i – 10j
48i – 42j

So now we have C but we want the direction. First it’s important to note which quadrant it will be in. The i direction is positive, so we know it will be right of the y axis, and the j term is negative so we know it will be below the x axis. this is quadrant 4, so we know the answer will be between 270 and 360

It is worth mentioning that arctan only gives aswers from -90 to 90 rather than a full 360,so we should be fine here, the answer will be between -90 and 0 when we take the arctan because that is quadrant 4 as well. if you has something like -5i + 3j you would expect the answer to be in quadrant 2, but you would get an answer in quadrant 4. the trick is to add 180. Similarly if you expect a result in quadrant 3 you will get the answer in quadrant 1. Same deal, just add 180. This is shown in Example 3 in the article

When you want the direction of a vector you take the i term as a and j term as b then take arctan(b/a)

here a = 48 and b = -42 so arctan(-42/48) = -41.19. just to double check this is in quadrant 4 so it is the right answer. If you started with a = -48 and b = 42 you would get the same answer but -48i + 42j is in quadrant 2, so you would take -41.19 and add 180 to get 138.81, which is in quadrant 2.

Let me know if this didn’t help

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