Let ${a} = \begin{pmatrix} 5 \\ -3 \\ -4 \end{pmatrix} \quad \text{and} \quad {b} = \begin{pmatrix} -11 \\ 1 \\ 28 \end{pmatrix}.$
There exist vectors ${p}$ and ${d}$ such that the line containing ${a}$ and ${b}$ can be expressed in the form
${v} = {p} + {d} t.$
Furthermore, for a certain choice of $ d$, it is the case that for all points ${v}$ lying on the same side of ${a}$ that ${b}$ lies on, the distance between ${v}$ and ${a}$ is $t$. Find ${d}$.
I'm not sure how to do problem. I tried choosing random points on the line, solving for $t$, and then solving the resulting system of equations to find $d$, but that's really messy. Any help is appreciated!
Hint: You want to start from the point $\vec{a}$ and the equation is $$\vec{v}=\vec{a}+t\vec{d}$$ where $\vec{d}$ is the direction vector which pointing from $\vec{a}$ to $\vec{b}$, and its norm is $1$.