Equation of a plane when a normal to the plane and the distance of the plane from the origin are given (a) Vector equation of a plane in normal form (b) Cartesian equation of a plane in normal form

**Equation of a plane when a normal to the plane
and the distance of the plane from the origin are given**

**Theorem 6.15**

The equation of the plane at a distance *p *from the origin
and perpendicular to the unit normal vector *d*Ë† is â‹… *d*Ë† = *p *.

**Proof**

Consider a plane whose perpendicular distance from the origin is *p
*.

Let *A *be the foot of the perpendicular from to the plane.

Let *d*Ë†
be the unit normal vector in the direction of .

Then = *pd*Ë† .

If * *is the position vector of an arbitrary point
*P *on the plane,

then * *is perpendicular to .

The above equation is called the vector equation of the
plane in **normal form**.

Let *l*, *m*, *n *be the direction cosines of *d*Ë†.
Then we have *d*Ë† = *li*Ë† + *m*Ë†*j *+ *nk*Ë†.

Thus, equation (1) becomes

. (*li*Ë† + *m*Ë†*j *+ *nk*Ë†) = p

If *P *is (*x,y,z*), then = *x*Ë†*i *+ *y*Ë†*j *+ *z*Ë†*k*

Therefore, (*xi*Ë† + *y*Ë†*j *+ *zk*Ë†) â‹…
(*li*Ë† + *m*Ë†*j *+ *nk*Ë†) = *p *or *lx *+ *my *+
*nz *= *p ...*.........(2)

Equation (2) is called the Cartesian equation of the plane in **normal form**.

**Remark**

(i) If the plane passes through the origin, then p = 0 . So, the
equation of the plane is *lx + my + nz*
= 0.

(ii) If is normal
vector to the plane, then Ë†*d* = is a
unit normal to the plane. So, the vector equation of the plane is . =
p or . = q , where q = p | | . The equation . = q is the
vector equation of a plane in **standard
form**.

**Note**

In the standard form . = q , need not be a
unit normal and q need not be the perpendicular distance.

Tags : Definition, Theorem, Proof , 12th Mathematics : UNIT 6 : Applications of Vector Algebra

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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Equation of a plane when a normal to the plane | Definition, Theorem, Proof

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