Altitude of the parallelogram

Question

The points $A(4, -2)$, $B(7, 2)$, $C(0,9)$, and $D(-3,5)$ form a parallelogram. Find the length of the altitude of this parallelogram with respect to the base $AB$.

Answer 1

HINT

Find the area by the cross product

$$S=|\vec v_1 \times \vec v_2|=|\vec v_1||\vec v_2|\sin \theta$$

with

$$v_1=(x_1,y_1,z_1)\;,\;\;\;v_2=(x_2,y_2,z_2)$$

$$v_1\times v_2:=\begin{vmatrix}\vec i&\vec j&\vec k\\x_1&y_1&z_1\\x_2&y_2&z_2\end{vmatrix}=(y_1z_2-z_1y_2\,,\,x_2z_1-x_1z_2\,,\,x_1y_2-x_2y_1)$$

and then $$h=\frac{S}{|AB|}$$