The position at time t of a particle P is given by $\displaystyle r(t)=\sqrt{2}ti+\frac{1}{2}t^2j+\ln(t)k$ for t>0. Find the speed and the angle the particle came through the xy-plane.

when I thought about his question I took the derivative r'(t) but why do I need to take the distance |r'(t)| ? Does this have to do with the difference between speed and velocity?

For the second part the work is $\displaystyle \arccos \frac{r'(1) \cdot k}{|r'(1)|\dot |k|}$ I know that the particle crosses the xy plane at t=1 so I get that part, but I'm lost as to how there's a triangle happening.

so the numerator has a dot product where the denominator has regular multiplication