Vector Subtraction

Question

Let [A] = 4î − 5ĵ and [B] = −9î+ 6ĵ What is [C] = [A] − 3 [B)]? Give your answer as a magnitude and direciton.

I'm able to get the magnitude, but whatever I find for the angle seems to be wrong.

Answer 1

If [B] = -9i + 6j then 3[B] = -27i + 18j

Then [A] - 3[B] = (4i - 5j) - (-27i + 18j) = 4i + 27i - 5j - 18j = 31i - 23j

The magnitude is √[(31)² + (-23)²] = 38.6

Here is a sketch:

The problem with the angle is what you are measuring from. To calculate θ from the positive x-axis as I have shown it in the sketch, we do θ = tan^-1(23/31) = 36.57° but since it is below the x-axis, I would call it -36.57°

If you are calculating the angle as a bearing (so many degrees off "north") then it would be 90° + 36.57° = 126.57°.

If you are using compass direction you might say it is S53.43°E (90 - 36.57 = 53-43)