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Vectors: Vector Plane Equation using the Normal
A plane through the point P( –2, 1, –5)
with normal n =
(
–6
–8
2
)
has the cartesian equation:
x + y + z + = 0

tickcross


Worked Solution:

If R(x, y, z) is a point in the plane then the vector
->
PR
 = r – p =
(
x
y
z
)
 –
(
–2
1
–5
)
 is perpendicular to n =
(
–6
–8
2
)
=>
->
PR
.n = 0
<=>(r – p) . n = 0
<=>r.n – p.n = 0
=> –6 x – 8 y + 2 z – ((–6)×(–2) + (–8)×1 + 2×(–5) = 0
<=> –6 x – 8 y + 2 z + 6 = 0
or any positive or negative multiple of this equation.

© MEI Produced by Dr Ron Knott, 20 may 2004
tom . button [AT] mei . org . uk 		Test reference: VectorsPlaneNorm.html?ref=78117