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# Image of a Point in a Plane

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  = 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 ˆi + 2ˆj + 3ˆk in the plane (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 Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
12th Mathematics : UNIT 6 : Applications of Vector Algebra : Image of a Point in a Plane |