Answer to Question #240794 in Analytic Geometry for Sakshi

Question

Let P QR be a triangle with coordinates P = (x1, y1, z1), Q = (x2, y2, z2) and R = (x3, y3, z3).

a) Prove that the triangle P QR lies on a plane. Find the equation of this plane.

b) Let (x, y, z) be a point in the plane containing the triangle. Prove that the det

x − x1 y − y1 z − z1

x − x2 y − y2 z − z2

x − x3 y − y3 z − z3

= 0.

Answer 1

a) Find the equation of a plane passing through three non-collinear points $(x_1 , y_1 , z _1), (x_2 , y_2 , z_ 2)$ and $(x_3 , y_3 , z _3).$

$$ax_1+by_1+cz_1=d$$ $$ax_2+by_2+cz_2=d$$ $$ax_3+by_3+cz_3=d$$

The system has unique solution if and only if

$$\begin{vmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \\ \end{vmatrix}\not=0$$

(b)

Let $(x, y, z)$ be a point in the plane containing the triangle. Then

$$ax+by+cz=d$$ $$ax_1+by_1+cz_1=d$$ $$ax_2+by_2+cz_2=d$$ $$ax_3+by_3+cz_3=d$$

Then

$$a(x-x_1)+b(y-y_1)+c(z-z_1)=0$$ $$a(x_2-x_1)+b(y_2-y_1)+c(z_2-z_1)=0$$ $$a(x_3-x_1)+b(y_3-y_1)+c(z_3-z_1)=0$$

System has nonzero solutions for $(a,b,c)$ if

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