Rigid body motion/ex-viii/R.M. Khan page 58

 Invariants under Orthogonal Transformation

 Orthogonal Transformation

We consider here three types of transformation of axes or transformation of coor dinates, namely 

1. translation 
2. rotation 
3. translation and rotation.

 These are called orthogonal transformation when both the systems of axes are rectangular. The combination of translation and rotation is called a rigid body motion.

 Invariants

Some expressions remain unchanged under an orthogonal transformation. These known as invariants of orthogonal transformation.

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                    Exercises-viii

1. The origin is shifted to the point (3.-3) without changing the directions of axes. If the coordinates of P, Q, R are (5,5), (-2, 4) and (7.-7) respectively in the new system, find the coordinates of these points in the old system. 

2. The axes are rotated through an angle of 60° without change of origin. The coordinates of a point are (4, √3) in the new system. Find the coordinates of it in the old system.

3.The origin is shifted to the point (3,-1) and the axes are rotated through an angle tan^-(3/4) If the coordinates of a point are (5, 10) in the new system, find the coordinates in the old system.

4. The coordinates of A and B are (5, -1) and (3,1). The origin is shifted to A and the axes are rotated in such a way that the new x-axis coincides with AB. If the rotation is made in the positive direction, find the formulae for transformation.

5. If ax + by and cx + dy are changed to a'x' + b'y' and c'x' + d'y' respectively for rotation of axes, show that ad - bc = a'd'- b'c'. (BH 911 NH 2007] 

6. Show that the radius of a circle remains unchanged due to any rigid body motion .

7. Show that there is only one point whose coordinates do not alter due to a rigid motion.

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