A man is standing on the roof of a building with his head at the position$\mathbf{<}\mathbf{12}\mathbf{,}\mathbf{30}\mathbf{,}\mathbf{13}\mathbf{>}$m. He sees the top of a tree, which is at the position $\mathbf{<}\mathbf{-}\mathbf{25}\mathbf{,}\mathbf{35}\mathbf{,}\mathbf{43}\mathbf{>}$m. (a) What is the relative position vector that points from the man's head to the top of the tree? (b) What is the distance from the man's head to the top of the tree?

Head position of man

Position of top of the tree m

(a) The tail of the position vector will be on the man's head (that has the position m where the tail of that vector is on the top of the tree (that has the position m). Now, to find the relative position vector between the two points subtract the position of the tail of the vector from the position of the head of that vector.

m.

Thus, the required relative position ism.

(b) To find the distance from the man's head to the top of the tree we find the magnitude of the position vector we found in part $\left(a\right)$of this problem (which is m).

$\left|\overrightarrow{\mathrm{r}}\right|=\sqrt{{\left(-37\right)}^2+{\left(5\right)}^2+{\left(30\right)}^2}=47.9\mathrm{m}$

Therefore, the required distance is$47.9$m.