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′ of A in the plane. Then  is perpendicular to the plane. So it is parallel to . Then

Let M be the middle point of AA′. Then the position vector of M is . But M lies on the plane.

Note

The mid point of M of AA′ 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 .

The coordinates of the image of a point in a plane

Let (a₁, a₂, a ₃) be the point  whose image in the plane is required. Then  = a₁ˆi + a2ˆj + a3ˆk 

Let ax + by + cz = d be the equation of the given plane. Writing the equation in the vector form we get . = p where  = aˆi + bˆj + cˆk Then the position vector of the image is

Example 6.55

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

Solution

Therefore, the image of the point with position vector iˆ + 2 ˆj + 3kˆ is 2iˆ + 4 ˆj + 7kˆ .

Note

The foot of the perpendicular from the point with position vector iˆ + 2ˆj + 3kˆ in the given plane is