Question 1197954


What is the angle {{{theta}}} between the two vectors {{{u}}}=<{{{3}}},{{{4}}}> and {{{v}}}<{{{2}}},{{{1}}}>

The angle {{{theta}}} between two vectors  {{{u }}}and {{{v}}} is related to the modulus (or magnitude) and scaler (or dot) product of  {{{u }}}and {{{v}}} by the relationship:


{{{u*v= abs(u)*abs(v)*cos(theta)}}}

first find dot product:
 
{{{u*v}}}=<{{{3}}},{{{4}}}>*<{{{2}}},{{{1}}}>

{{{u*v=3*2 +4*1}}}

{{{u*v=10}}}


now find the modulus: 


{{{abs(u)=sqrt(3^2+4^2)=sqrt(25)=5}}}

{{{abs(v)=sqrt(2^2+1^2)=sqrt(5)}}}

their product:

{{{abs(u)*abs(v)=5sqrt(5)}}}


plug all in formula above:

{{{10= 5sqrt(5)*cos(theta)}}}

{{{10/(5sqrt(5))=cos(theta)}}}

{{{2/sqrt(5)=cos(theta)}}}

{{{theta=cos^-1(2/sqrt(5))}}}

{{{theta=0.463647609}}}-> radians

{{{theta=26.57}}}°