Finding the length of a 3-dimensional vector

Question

I am struggling with a Linear Algebra problem that involves finding the length of a 3-dimensional vector $\mathbf r$, as shown in the picture I sketched:

I do not have the coordinates of the points in this case, but for example, I know that the length of the vector $\mathbf v$ is:

$$||\mathbf v||=\sqrt{x^2+y^2+z^2}$$

Is there any similar way to find the length (in respect to $x$, $y$ and $z$) of the vector $\mathbf r$ in this case? If so, could anyone please explain me?

Answer 1

It depends what point on the $Z$-axis r ends on. Assuming you want the shortest r possible:

r is shortest when it is perpendicular to the $Z$-axis ends

$\implies$ r ends at $(0, 0, z)$

Note: that since r goes from $(x, y, z)$ to $(0, 0, z)$ it is parallel to the $xy$-plane

$\implies$ the length of r is the same as the length $(x, y, 0)$ to $(0, 0, 0)$

So,

||r|| = $\sqrt{ x^2 + y^2}$