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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**

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**

If *a*, *b*, *c *are the direction ratios of ,
then we have = *ai*Ë† + *b*Ë†*j *+ *ck*Ë†.

Suppose, *A *is (*x*_{1} , *y*_{1}
, *z*_{1}) then equation (1) becomes ((*x *âˆ’ *x*_{1}
)*i*Ë† + ( *y *âˆ’ *y*_{1} ) Ë†*j *+ (*z *âˆ’ *z*_{1}
)*k*Ë†) â‹… (*ai*Ë† + *b*Ë†*j *+ *ck*Ë†) =
0 . That is,* *

*a*(*x *âˆ’ *x*_{1}) + *b*( *y *âˆ’ *y*_{1})
+ *c*(*z *âˆ’ *z*_{1}) = 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 (*x*_{1} , *y*_{1} , *z*_{1}) .

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

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

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