Question

I need help with Q3 point 2 please

Answer 1

Take the given vector by itself, $\mathbf v=2\,\mathbf i+\mathbf j-\mathbf k$. Scaling the vector by a scalar $t$ will stretch the vector. Any real number can take the place of $t$; when we consider all possible choices, we see that $t\mathbf v$ traces out an infinite line through the origin and the point (2, 1, -1). (Left)

Next, treat point P(2, 1, -3) as a vector, $\mathbf P=2\,\mathbf i+\mathbf j-3\,\mathbf k$. (Upper right) By adding $\mathbf P$ to $t\mathbf v$, we get a new vector that, as we change the value of $t$, traces out the line through point P which also happens to be parallel to $\mathbf v$. (Lower right)

So the equation of this line in parametric form is

$\mathbf r(t)=(2,1,-3)+t(2,1,-1)=(2+2t,1+t,-3-t)$

or in ijk notation,

$\mathbf r(t)=(2+2t)\,\mathbf i+(1+t)\,\mathbf j-(3+t)\,\mathbf k$