Question

Three vectors A, B and C are such that

A = B + C and their magnitudes are 5, 4 and 3 respec-
tively. Find the angle between A and C.

Please answer with explanation

Answer 1

Since the magnitudes satisfy the identity for right triangles

`A^2=B^2+C^2`

we deduce that B and C are perpendicular, and that ABC is a right triangle (see picture).

So, we can get the angle between A and C using the sine theorem:

`\frac{A}{\sin(\hat{A})}=\frac{B}{\sin(\hat{B})}`

And we have

`A=5,\quad B=4,\quad \hat{A}=90`

So, we have

`\sin(\hat{B})=\frac{B\sin(\hat{A})}{A}=(4\sin(90))/(5)=(4)/(5)`

Which implies

`\hat{B}=\arcsin\left((4)/(5)\right)\approx 53.13`