Transformation of Axes
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Transcript Transformation of Axes
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 3x2 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
13x2 13 y 2 10z 2 8xy 4 yz 4zx 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.