I have done this TNB problem multiple times; however, my online homework system keeps telling me my answer is incorrect. I was hoping someone would look at my work and tell me where I'm going wrong?
The problem gives me the position vector $\vec{r}(t) = \langle 8\cos(t) , 17\sin(t) , 15\cos(t)\rangle$ and I need to find the normal vector $\vec{N}(\pi/4)$.
Here is my work
Let's write $$\vec{r}(t) = \langle 8,0,15 \rangle \cos(t) + \langle 0,17,0 \rangle \sin(t).$$ Since the vectors $\langle 8,0,15 \rangle$ and $\langle 0,17,0 \rangle$ are perpendicular with common length 17, $\vec{r}$ traces out a circle centered at the origin. Something like so:
The vector $\vec{N}(\pi/4)$ is perpendicular to this circle, contained in the plane of the circle, points back towards the origin, and has unit length. Thus, it must be $$\vec{N}(\pi/4) = -\frac{\vec{r}(\pi/4)}{||\vec{r}(\pi/4)||} = \left\langle-\frac{4 \sqrt{2}}{17},-\frac{1}{\sqrt{2}},-\frac{15}{17 \sqrt{2}}\right\rangle.$$
Note that this agrees with your computation. Perhaps your online homework system expects an exact answer.