Question

Let the plane P pass through the 3 points R = (1, 1, 6), S= (2,5,4) and T = (1,2,3).

Find the values of a, b, c and d in the equation $ax + by + cz = d$ of plane P.

Answer 1

The plane through three points

$$\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \\ \end{vmatrix}=0$$

$$\begin{vmatrix} x-1 & y-1 & z-6 \\ 2-1 & 5-1 & 4-6 \\ 1-1 & 2-1 & 3-6 \\ \end{vmatrix}=0$$

$$(x-1)\begin{vmatrix} 4 & -2 \\ 1 & -3 \end{vmatrix}-(y-1)\begin{vmatrix} 1 & -2 \\ 0 & -3 \end{vmatrix}+(z-6)\begin{vmatrix} 1 & 4 \\ 0 & 1 \end{vmatrix}=0$$

$$-10(x-1)+3(y-1)+z-6=0$$

$$10x-3y-z=1$$

$$a=10, b=-3, c=-1, d=1$$