Give a Cartesian for planes through given points spanned by given vectors

Consider the vectors

$A = 2 \mathbf{i} + 3 \mathbf{j} - 4 \mathbf{k}, \qquad B = \mathbf{j} + \mathbf{k}.$

Find a nonzero vector $N$ perpendicular to both $A$ and $B$ .
Find a Cartesian equation for the plane through $(0,0,0)$ which is spanned by $A$ and $B$ .
Find a Cartesian equation for the plane through $(1,2,3)$ which is spanned by $A$ and $B$ .

Since $A$ and $B$ are independent, we can take
$N = A \times B = (3 - (-4), 0 - 2, 2 - 0) = (7,-2,2).$
From part (a) we have $N = (7,-2,2)$ is perpendicular to both $A$ and $B$ , so a Cartesian equation for the plane is given by
$7x - 2y + 2z = d$

Further, since the point $(0,0,0)$ is on the plane, we must have $d = 0$ . Hence, the Cartesian equation for the plane is

$7x - 2y + 2z = 0.$
Again, we have a Cartesian equation for the plane given by
$7x - 2y + 2z = d.$

Since $(1,2,3)$ is on the plane we must have

$d = 7(1) - 2(2) + 2(3) \quad \implies \quad d = 9.$

Hence, the Cartesian equation for the plane is given by

$7x - 2y + 2z = 9.$