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School of Mathematics and Statistics, The University of Sydney
 13. Planes in space
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The Cartesian equation of a plane

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Suppose that P0 has coordinates x0,y0,z0 and n has components a,b,c.

    n                        n    P0    r - r0                  P     r0        r       O

Let P, with coordinates x,y,z, be an arbitrary point on the plane.

The position vectors of P0 and P are

-O-P--> =  r =  x i + y j + z k    0    0    0     0    0

and

-OP-->  = r = xi + yj + zk.

Substituting

 r = xi + yj + zk,  r0 = x0i + y0j + z0k  n = ai + bj + ck

into the vector equation, we obtain

(                                )   (x-  x0)i + (y - y0)j + (z - z0)k · (ai + bj + ck) = 0

which, when multiplied out, gives

a(x - x0) + b(y - y0) + c(z-  z0) = 0.

This is called a Cartesian equation of the plane.

It simplifies to

ax + by + cz = d

where d is the constant ax0 + by0 + cz0.

An equation of the form

ax + by + cz = d

where a,b,c and d are constants and not all a,b,c are zero, can be taken to be an equation of a plane in space.

The coefficients a, b and c are the components of a normal vector

n =  ai + bj + ck

for the plane described by the equation.

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