If #A = {2,-7 ,5}#, #B = {5,-7,4}# and #C = A - B#, what is the angle between A and C?

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Answer 1 · 2016-04-05T11:01:31

#alpha=90,02 ^o#

#"a) find C=A-B"#

#C=(A_x-B_x),(A_y-B_y),(A_z-B_z)#

#C=(2-5),(-7+7),(5-4)#

#C={-3,0,1}#

#"b) find the magnitude of A and C:"#

#||A||=sqrt(A_x^2+A_y^2+A_z^2)#

#||A||=sqrt(2^2+(-7)^2+5^2)=sqrt(4+49+25)=sqrt78#

#||C||=sqrt(C_x^2+C_y^2+C_z^2)#

#||C||=sqrt((-3)^2+0^2+1^2)=sqrt(9+0+1)=sqrt10#

#"c) find dot product of A.C" #

#A*C=A_x*C_x+A_y*C_y+A_z*C_z#

#A*C=2*(-3)+(-7)*0+5*1#

#A*C=-6+0+5=-1#

#"d) use the dot product formula"#

#A.C=||A||*||C||*cos alpha#

#A*C=-1#
#||A||=sqrt78#
#||C||=sqrt10#

#-1=sqrt78*sqrt10*cos alpha#

#cos alpha=-1/(sqrt78*sqrt10)#

#cos alpha=-1/sqrt780#

#cos alpha=-1/(27,93)#

#alpha=90,02 ^o#