Adding vectors - finding vector B with resultant and vector A

Question

Hello,

I've been trying to figure out the following question for some time : the resultant of two vectors, A and B, is a vector of 40 meters towards the north. If the magnitude of A is 30 m and it forms a 30 degree angle towards the south in relation to the west, determine the vector B.

Thank you.

Answer 1

I believe this sketch matches your problem;

Split vector A into its components:

A_x = 30cos(30°) = 25.98 (going west)

A_y = 30sin(30°) = 15 (going south)

Since the resultant has no x-component (it's straight up toward north), then B_x must be the same magnitude but opposite direction to A_x so B_x = 25.98 (east).

For the y-direction, A_y + B_y must equal the Resultant (since it is only in the y-direction). Since Ay is going south, we can call its value -15. So -15 + B_y = 40 or B_y = 55

Since Bx = 25.98 and By = 55, the magnitude of B = √(25.98² + 55²) = √3700 = 60.83 m.

To find the direction, use tan(θ) = B_x/B_y = 25.98/55 so θ = tan^-1(25.98/55) = 25.3° west of north