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Chapter: 12th Mathematics : UNIT 6 : Applications of Vector Algebra

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



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

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12th Mathematics : UNIT 6 : Applications of Vector Algebra


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