The coordinates of the image of a point in a plane

**Image
of a Point in a Plane**

Let A be the given point whose position vector is .
Let â‹… * *= *p *be the equation
of the plane.

Let * *be the position vector of the mirror
image *A*â€² of *A *in the plane. Then is
perpendicular to the plane. So it is parallel to . Then

Let M be the middle point of AAâ€². Then the position vector of M is . But M lies on the plane.

The mid point of *M*
of *AA*â€² is the foot of the
perpendicular from the point *A* to the
plane . = p.

So the position vector of the foot *M* of the perpendicular is given by .

Let (*a*_{1},
*a*_{2}, *a *_{3}) be the point * *whose
image in the plane is required. Then = *a*_{1}*Ë†i *+ *a*2*Ë†j *+ *a*3*Ë†k *

Let *ax *+ *by *+ *cz *= *d *be the equation
of the given plane. Writing the equation in the vector form we get . = p* *where = *aË†i *+ *bË†j
*+ *cË†k *Then the position vector of the image is

Find the image of the point whose position vector is *Ë†**i *+ 2*Ë†**j *+ 3*Ë†**k* in
the plane â‹… (*Ë†**i *+ 2*Ë†**j *+ 4*Ë†**k *) = 38 .

Therefore, the image of
the point with position vector *i*Ë† + 2
Ë†*j *+ 3*k*Ë† is 2*i*Ë† + 4 Ë†*j *+ 7*k*Ë†
.

**Note**

The foot of the
perpendicular from the point with position vector *i*Ë† + 2Ë†*j *+ 3*k*Ë† in the given plane is

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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Image of a Point in a Plane |

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