Transformation of Axes Change of origin To change the origin of co-ordinate without changing the direction. Change of Direction Of Axes(without changing origin) Let direction cosines of new axes O’X’, O’Y’,O’Z’ through O be . To find X: Multiply elements of x-row i.e. By x’,y’,z’ and add. Similarly for y and Z

To find element of x’ column i. e. , by x, y, z and add To find element of x’ column i.e., by x, y, z and add. Similarly for y’ and z’ Note: The degree of an equation remains unchanged, if the axes are changed without changing origin, because the transformation are linear. Art3. Relation between the direction cosines of three mutually perpendicular lines. Art4. If be the direction cosine of three mutually perpendicular lines, then

Example1. Find the co-ordinate of the points (4,5,6) referred to parallel axes through (1,0,-1). Example2. Find the equation of the plane 2x+3y+4z=7 referred to the point (2,-3,4) as origin, direction of axes remaining same. Example3. Reduce to form in which first degree terms are absent. Example4:Transform the equation When the axes are rotated to the position having direction cosine.

<-1/3,2/3,2/3>,<2/3,-1/3,2/3>,<2/3,2/3,-13> Example: If are direction cosines of three mutually perpendicular lines OA=OB=OC=a, Prove that the equation of plane ABC is Example: If be direction cosines of the three mutually perpendicular lines, prove that the line whose direction cosines are proportional to makes equal angles with them.