Question

What displacement must be added to a 50cm displacement in the +x-direction to give a resultant displacement of 85cm at 25 degree's

Answer 1

Let denote the initial displacement with a vector $\mathbf{d}_1$ with the following coordinates:

$$\mathbf{d}_1 = (50, 0)$$

and the unknown displacement with $\mathbf{d}_2 = (x,y)$.

The final displacement is:

$$\mathbf{d} = 85\cdot (\cos25\degree, \sin25\degree)$$

Then, since

$$\mathbf{d}_1 + \mathbf{d}_2 = \mathbf{d}\\ \mathbf{d}_2 = \mathbf{d}-\mathbf{d}_1\\ (x, y) = 85\cdot (\cos25\degree, \sin25\degree) - (50, 0)$$

Comparing coordinatewisely both sides, find:

$$x = 85\cos25\degree-50\approx 27cm\\ y = 85\sin25\degree-0 \approx 36cm$$

Answer. (27, 36) cm.