Question #240069

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.


1
Expert's answer
2021-09-21T18:01:09-0400

a) Find the equation of a plane passing through three non-collinear points (x1,y1,z1),(x2,y2,z2)(x_1 , y_1 , z _1), (x_2 , y_2 , z_ 2) and (x3,y3,z3).(x_3 , y_3 , z _3).


ax1+by1+cz1=dax_1+by_1+cz_1=d

ax2+by2+cz2=dax_2+by_2+cz_2=d

ax3+by3+cz3=dax_3+by_3+cz_3=d

The system has unique solution if and only if


x1y1z1x2y2z2x3y3z30\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)(x, y, z) be a point in the plane containing the triangle. Then


ax+by+cz=dax+by+cz=d


ax1+by1+cz1=dax_1+by_1+cz_1=d

ax2+by2+cz2=dax_2+by_2+cz_2=d

ax3+by3+cz3=dax_3+by_3+cz_3=d

Then


a(xx1)+b(yy1)+c(zz1)=0a(x-x_1)+b(y-y_1)+c(z-z_1)=0

a(x2x1)+b(y2y1)+c(z2z1)=0a(x_2-x_1)+b(y_2-y_1)+c(z_2-z_1)=0

a(x3x1)+b(y3y1)+c(z3z1)=0a(x_3-x_1)+b(y_3-y_1)+c(z_3-z_1)=0


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


D=xx1yy1zz1x2x1y2y1z2z1x3x1y3y1z3z1=0D=\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


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