[ cos a - sin a 07 cosB EXAMPLE 9 L...
Question

# [ cos a - sin a 07 cosB EXAMPLE 9 Let Fla) = sin a cos a 0 and G(B) = 0 | 0 0 1 L-sin B Show that [F(a). GBT? = G(+B) · F1-a). 0 1 0 sin B] 0 cos B]

11th - 12th Class
Maths
Solution
110
4.0 (1 ratings)
We have ( begin{aligned} F(alpha) cdot F(-alpha)=left[begin{array}{ccc}cos alpha & -sin alpha & 0 sin alpha & cos alpha & 0 0 & 0 & 1end{array}right]left[begin{array}{ccc}cos (-alpha) & -sin (-alpha) & 0 sin (-alpha) & cos (-alpha) & 0 0 & 0 & 1end{array}right] =&left[begin{array}{ccc}cos alpha & -sin alpha & 0 sin alpha & cos alpha & 0 0 & 0 & 1end{array}right]left[begin{array}{ccc}cos alpha & sin alpha & 0 -sin alpha & cos alpha & 0 0 & 0 & 1end{array}right] operatorname{Thus}, F(alpha) cdot F(-alpha)=I Rightarrow{F(alpha)}^{-1}=F(-alpha) & 0 sin i tan mid y, G(beta) cdot G(-beta)=I & Rightarrow{G(beta)}^{-1}=G(-beta) end{aligned} ) ( left.therefore quad{F(alpha) cdot G(beta)}^{-1}={G(beta)}^{-1} cdot{F(alpha)}^{-1} quad text { [by reversal law }right] ) [ =G(-beta) cdot F(-alpha) ] Hence, ( {F(alpha) cdot G(beta)}^{-1}=G(-beta) cdot F(-alpha) )