|a+b|=30N, |b|=24N and angle between(a+b) and b is 53°. Find a)the angle between a and b b)|a| and |a-b| c) angle between a and (a+b)?

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Answer 1 · 2018-06-21T06:23:48

Please see the explanation below.

Apply the cosine rule to calculate #|a|#

#a^2=b^2+c^2-2bc cosA#

#|a|^2=|a+b|^2+|b|^2-2*|a+b|*|b|*cos53#

#|a|^2=30^2+24^2-2*30*24*cos(53)#

#=609.4#

#|a|=sqrt(609.4)=24.7#

Apply the sine rule to calculate the angle between #|a|# and #|b|#

#sin(53)/24.7=sin(180-alpha)/30#

#sin(180-alpha)=30/24.7*sin53=0.97#

#180-alpha=76.1#

#alpha=180-76.1=103.9^@#

#(a)# The angle between #|a|# and #|b|# is #=103.9^@#

#(c)#The angle between #|a|# and #|a+b|# is #=103.9-53=50.9^@#

#(b)# Calculate #|a-b|#

#|a-b|^2=|a|^2+|b|^2-2*|a|*|b|*cos(103.9)#

#|a-2|^2=24.7^2+24^2-2*24.7*24*cos103.9#

#=1470.9#

#|a-b|=sqrt(1470.9)=38.4#

Apply the sine rule to calculate the angle between #|a|# and #|a-b|#

#24/sinbeta=38.4/sin103.9#

#sinbeta=24/38.4*sin103.9=0.61#

#beta=37.4^@#