The image I'm including IS from a mymathlab assignment, but it's a practice assignment, so it's not homework. I just need to understand how it was done.
I understand the concept of calculating v(t). You take the derivative of each individual component and them put them back in ijk form. That makes sense to me, and I got the first step correct. The next part is where I get confused.
On the very next step, they show v as equal to (18t^2)i - (2t^2)j + (3t^2)k.
This confuses me because in the very previous step we had established that v was actually equal to (18t^2)i + (-6t^2)j + (9t^2)k. Why in the next step are the j and k components suddenly different?
Finally, I don't know how to calculate the magnitude from even those components. I would set it up as sqrt((18t^2)i - (2t^2)j + (3t^2)k) but I don't know how to calculate that square root.
Please help
You can pull the $t^2$ factor $$ v = 18 t^2 i - 2t^2 j + 3t^2 k = t^2 (18i-2j+3k) $$ and then using $\lVert \alpha x \rVert = \lvert \alpha \rvert \lVert x \rVert$: $$ \lVert v \rVert = \lvert t^2 \rvert \lVert 18i-2j+3k \rVert = t^2 \sqrt{18^2 + 2^2 + 3^2} = t^2 \sqrt{337} = t^2 \cdot 18.35\dotsb $$ If we try the other version: $$ \lVert v \rVert = \lvert t^2 \rvert \lVert 18i-6j+9k \rVert = t^2 \sqrt{18^2 + 6^2 + 9^2} = t^2 \sqrt{441} = t^2 \cdot 21 $$ This fits better. So it seems to be some error which slipped in.