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# Equation of a plane perpendicular to a vector and passing through a given point

Equation of a plane perpendicular to a vector and passing through a given point

Equation of a plane perpendicular to a vector and passing through a given point

## (a) Vector form of equation

Consider a plane passing through a point A with position vector and is a normal vector to the given plane.

Let be the position vector of an arbitrary point P on the plane.

Then is perpendicular to . which is the vector form of the equation of a plane passing through a point with position vector and perpendicular to .

Note ## (b) Cartesian form of equation

If a, b, c are the direction ratios of , then we have = aiˆ + bˆckˆ.

Suppose, A is (x1 , y1 , z1) then equation (1) becomes ((x x1 )iˆ + ( y y1 ) ˆj + (z z1 )kˆ) (aiˆ + bˆj + ckˆ) = 0 . That is,

a(x x1) + b( y y1) + c(z z1) = 0

which is the Cartesian equation of a plane, normal to a vector with direction ratios a, b, c and passing through a given point (x1 , y1 , z1) .

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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Equation of a plane perpendicular to a vector and passing through a given point |