Find the vector equation of a line which contains the two points $(1, 2, 3)$ and $(4, 5, 6) $
Hints only.
I know $x = a + t*d$ where $a, d$ are vectors and $t$ is a scalar.
We can have $a =\{<1, 2, 3>, <4, 5, 6> \}$ as two options of the vectors.
The difficulty is in finding $d$?
Hint. You want both $u := (1,2,3)$ and $v: = (4,5,6)$ to lie on the line, that is for some values of $t_u, t_v$, we want $$ a + t_ud = u,\qquad a + t_vd = v $$ as we are free in choosing $t_u$ and $t_v$ (we just want them to lie on the line $a + \mathbf R d$ somewhere, let $t_u = 0$ and $t_1 = 1$, this gives $$ a = u, \qquad a + d = v $$ Can you continue from here?