The Vector Equation Of A Plane

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13. Planes in space
Main menu Section menu Previous section Next section button disabled The Cartesian equation of a plane
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The vector equation of a plane

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A plane can be described in many ways. The plane, for example, can be specified by three non-collinear points of the plane: there is a unique plane containing a given set of three non-collinear points in space.

An alternative way to specify a plane is given as follows.

Select a point P0 in the plane. There is a unique line through P0 perpendicular to the plane. This line is called the normal to to the plane at P0. A vector n/=0 parallel to this normal is called a normal vector for the plane.

There is a unique plane which passes through P0 and has n as a normal vector.

n n P0 r - r0 P r0 r O

Now P lies in the plane through P0 perpendicular to n if and only if -P---P 0 and n are perpendicular.

As ---- P0P = r - r0, this condition is equivalent to

(r - r0) · n = 0.

This is a vector equation of the plane.

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