How do I find the angle θ between two vectors?

Question

My givens are this:
u = <3, 2>
v = <4, 0>

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Answer 1 · 2018-02-15T21:03:37

#theta=cos^-1(3/sqrt13)#

Given:
#vecu=3i+2j#
#vecv=4i+0j#

Consider the dot product
#vecu.vecv=|vecu|.|vecv|.costheta#
where #theta# represents the angle between the vectors #vecu and vecv#

#vecu.vecv=(3)(4)+(2)(0)=12+0=12#

#|vecu|=sqrt(3^2+2^2)=sqrt(9+4)=sqrt13#

#|vecv|=sqrt(4^2+0^2)=sqrt(16+0)=sqrt16=4#

Substituting

#12=(sqrt13)(4)costheta#

Interchanging lhs and rhs

#(sqrt13)(4)costheta=12#

#costheta=12/(4sqrt13)=3/sqrt13#

#costheta=3/sqrt13#

Hence,
#theta=cos^-1(3/sqrt13)#