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  l1, m1, n1 ,  l2 , m2 , n2 ,  l3 , m3 , n3  .
To find X: Multiply elements of x-row i.e.
x’,y’,z’ and add. Similarly for y and Z
l1 , l2 , l3
By
• To find element of x’ column i.e.,
and add. Similarly for y’ and z’
l1 , m1 , n1
by x, y, 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  l1, l2 , l3 ,  m1, m2 , m3 ,  n1, n2 , n3  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 3x 2  2 y 2  z 2  6 x  8 y  4 z  11
to form in which first degree terms are absent.
Example4:Transform the equation
13x 2  13 y 2  10 z 2  8 xy  4 yz  4 zx  144
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  l1, l2 , l3 ,  m1, m2 , m3 ,  n1, n2 , n3  are
direction cosines of three mutually perpendicular
lines OA=OB=OC=a, Prove that the equation of plane
ABC is
(l1  l2  l3 ) x  (m1  m2  m3 ) y  (n1  n2  n3 ) z  a
• Example: If  l1, l2 , l3 ,  m1, m2 , m3 ,  n1, n2 , n3  be direction
cosines of the three mutually perpendicular lines,
prove that the line whose direction cosines are
proportional to  l1  l2  l3  m1  m2  m3  n1  n2  n3 
makes equal angles with them.