If #A = <4 ,-5 ,-4 >#, #B = <-9 ,1 ,-6 ># and #C=A-B#, what is the angle between A and C?

1 Answer
Feb 26, 2016

#0.801 " radians" or 45.67^@#

Explanation:

If #A=< color(blue)(4, -5, -4) >#
and #B = < color(green)( -9, 1, -6) >#
then
#color(white)("XXX")C= A-B= < color(blue)4-(color(green)(-9)), color(blue)(-5)-color(green)(1), color(blue)(-4)-(color(green)(-6)) >#

#color(white)("XXXX")=< color(red)(13, -6, 2) >#

#theta_(AC) = arccos ((A*C)/(||A||xx||C||))#

#A*C= color(blue)(4)xxcolor(red)(13) +color(blue)((-5))xxcolor(red)((-6))+color(blue)((-4))xxcolor(red)(2)#
#color(white)("XXX")= 76#

#||A|| = sqrt(color(blue)(4)^2+color(blue)((-5))^2+color(blue)((-4))^2)#
#color(white)("XXX")=sqrt(57)#

#||C|| =sqrt(color(red)(13)^2+color(red)((-6))^2+color(red)(2)^2)#
#color(white)("XXX")=sqrt(209)#

#theta= arccos(76/(sqrt(57)*sqrt(209))) = arccos(4/sqrt(33))#

#color(white)("X")=0.801 " radians" or 45.67^@#