Calculate |#vec u# + #vec v#| help?

Question

Given that |#vec u#| = 3 and |#vec v#| = 4

ALSO
calculate the angle between #vec u# and
#vec u# + #vec v#

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Answer 1 · 2018-06-09T06:55:16

See below

First Part

#abs bb u = 3, abs bb v = 4, phi = (2pi)/3#

#abs(bb u + bb v)^2 = (bb u + bb v )*(bb u + bb v )#

#= u^2 + v^2 + 2 bb u * bb v#

#= 3^2 + 4^2 + 2 * 3 * 4 * cos(120)#

#=13#

#implies abs(bb u + bb v) = sqrt13#

Second Part

#bb u * (bb u + bbv) =abs bb u * abs (bb u + bbv) cos theta#

#cos theta = (bb u * (bb u + bbv))/ (abs bb u * abs (bb u + bbv) ) #

#= (u^2 +abs bb u abs bb v cos phi)/ (abs bb u * abs (bb u + bbv) ) #

#= (3^2 +3 * 4 * cos (120))/ (3 * sqrt(13) ) #

#= 1/sqrt(13)#

#theta = cos^(-1) (1/sqrt(13)) = 73.9^o#