Find the values of $ t$ so that the tangent line to the given curve contains the given point

Question

I'm doing exercise 27.

I know that the tangent vector is $ r'(t)$.

I think that the tangent line should be $ <-8,2,-1> + \frac{r'(t)}{ ||{r'(t)||}}$.

I'm not sure how to find all values of $ t$ . I suppose that $t \in \mathbb{R}$. I'm not sure what this question is asking.

I'm looking to verify my solution and correct if necessary.

Answer 1

Note that an equation for a line through point $\mathbf p$ with direction $\mathbf v$ is $\mathbf s=\mathbf p+\mathbf v t$. Also note that this $t$ is independent of the $t$ in the given function. Further, the magnitude of your direction vector $\mathbf v$ doesn't matter. So, you're looking for the values of $t$ such that $\langle -8,2,-1\rangle=\mathbf r(t)+c\mathbf r'(t)$ for some scalar $c$.