$a.$ Given coordinates $(x, y, z )$ with origin $(0,0,0)$, parameterize the line through the points $(4,5,6)$ and $(1,2,3).$
$b.$ Take components of your answer to Part $(a)$ to give three scalar equations parameterizing $l$: $x(t) = \ldots, y(t) = \ldots, z(t) = \ldots$
$a.$ Let $(x, y, z), (4, 5, 6), (1, 2, 3)$ be $3$ vectors all originating at $(0, 0, 0).$ Let $l$ be a line passing through the head of those $3$ vectors. Then by head-to-tail method, $(4, 5, 6) - (1, 2, 3)$ is a vector on $l$. Scaling it by $t$ gives us another vector $t((4, 5, 6) - (1, 2, 3))$ on $t$. Then by head-to-tail method, $(x, y, z) = t(4, 5, 6) + (1 - t)(1, 2, 3).$
$b.$ $x(t) = 4t + (1 - t) = 3t + 1.$
$y(t) = 5t + 2 - 2t = 3t + 2.$
$z(t) = 3t + 3.$
Does that make sense?