Page 99 - A Course in Linear Algebra with Applications

3.3: Determinants and Inverses of Matrices 83

therefore determine a unique plane. Find the equation of the
plane by using determinants.
We know from analytical geometry that the equation of
the plane must be of the form ax + by + cz + d = 0. Here
the constants a, b, c, d cannot all be zero. Let P(x,y,z) be
an arbitrary point in the plane. Then the coordinates of the
points P, Pi, P2, P3 must satisfy the equation of the plane.
Therefore the following equations hold:

ax + by + cz + d = 0
ax\ + byi + cz\ + d = 0
ax2 + bx 2 + cz 2 + d = 0
+ + + d = 0
ax 3 by 3 cz 3
Now this is a homogeneous linear system in the unknowns
a, b, c, d; by 3.3.2 the condition for there to be a non-trivial
solution is that
x y z 1
xi y x z\ 1
2/2 1
x 2 z 2
£3 2/3 z 3 1
This is the condition for the point P to lie in the plane, so it
is the equation of the plane. That it is of the form ax + by +
cz + d = 0 may be seen by expanding the determinant by row
1.
For example, the equation of the plane which is deter-
mined by the three points (0, 1, 1), (1, 0, 1) and (1, 1, 0)
is
x y z 1
0 1 1 1
1 0 1 1 '
1 1 0 1
which becomes on expansion x + y + z — 2 = 0.