If the matrixbegin mathsize 20px style R open parentheses alpha close parentheses equals open square brackets table row cell cos alpha end cell cell space minus sin alpha end cell cell space 0 end cell row cell sin alpha end cell cell space cos alpha end cell cell space 0 end cell row 0 0 cell space 1 end cell end table close square brackets comma t h e n space s h o w space t h a t space
R open parentheses alpha close parentheses. R open parentheses beta close parentheses equals R open parentheses alpha plus beta close parentheses. H e n c e space p r o v e space t h a t space open square brackets R open parentheses alpha close parentheses close square brackets to the power of negative 1 end exponent equals R open parentheses negative alpha close parentheses
end style

Asked by sunil2791 | 29th Jul, 2017, 10:48: AM

Expert Answer:

begin mathsize 16px style straight R left parenthesis straight beta right parenthesis equals open square brackets table row cosβ cell negative sinβ end cell 0 row sinβ cosβ 0 row 0 0 1 end table close square brackets space and space straight R left parenthesis straight alpha plus straight beta right parenthesis equals open square brackets table row cell cos open parentheses straight alpha plus straight beta close parentheses end cell cell negative sin open parentheses straight alpha plus straight beta close parentheses end cell 0 row cell sin open parentheses straight alpha plus straight beta close parentheses end cell cell cos left parenthesis straight alpha plus straight beta right parenthesis end cell 0 row 0 0 1 end table close square brackets
straight R left parenthesis straight alpha right parenthesis times straight R left parenthesis straight beta right parenthesis equals open square brackets table row cosα cell negative sinα end cell 0 row sinα cosα 0 row 0 0 1 end table close square brackets open square brackets table row cosβ cell negative sinβ end cell 0 row sinβ cosβ 0 row 0 0 1 end table close square brackets
After space multiplying space you space will space get space
open square brackets table row cell cos open parentheses straight alpha plus straight beta close parentheses end cell cell negative sin open parentheses straight alpha plus straight beta close parentheses end cell 0 row cell sin open parentheses straight alpha plus straight beta close parentheses end cell cell cos left parenthesis straight alpha plus straight beta right parenthesis end cell 0 row 0 0 1 end table close square brackets space
equals straight R open parentheses straight alpha plus straight beta close parentheses
using space cos left parenthesis straight x plus straight y right parenthesis equals cosxcosy minus sinxsiny
Using space straight R left parenthesis straight alpha right parenthesis times straight R left parenthesis straight beta right parenthesis equals straight R open parentheses straight alpha plus straight beta close parentheses
Multiplying space by space straight R to the power of negative 1 end exponent open parentheses straight alpha close parentheses space both space sides space by space left space side comma
yo u space can space find space open square brackets straight R left parenthesis straight alpha right parenthesis close square brackets to the power of negative 1 end exponent.



you space can space prove space it space equals space to space straight R left parenthesis negative straight alpha right parenthesis end style

Answered by Sneha shidid | 31st Jul, 2017, 10:07: AM