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`