Question 1206880

{{{u = i + 3j}}}, {{{v = -3i + j}}}

(a) Find {{{u * v}}}


To find the dot product of {{{u}}} and {{{v}}}, use the formula 

{{{u * v= (a[1] *a[2]) + (b[1] *b[2]) }}}

where {{{a[1]}}}, {{{a[2]}}} are the coefficients of {{{i}}} and {{{b[1]}}},{{{b[2]}}} are the coefficients of {{{j}}}.


 In this case, 

{{{a[1]=1}}}, {{{b[1]=3}}}
{{{a[2]=-3}}}, {{{b[2]=1}}}


the dot product of {{{u }}}and {{{v}}} is 

{{{u *v = (1 *( -3)) + (3 *1) =-3+3=0}}}



(b) Find the angle between {{{u }}}and {{{v}}}  to the nearest degree.

To find the angle between {{{u }}}and {{{v}}} , we use the formula:


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


where {{{abs(u)}}} and {{{abs(v) }}}are the magnitudes of {{{u }}}and {{{v}}} , respectively

the magnitude of {{{u}}} is {{{sqrt(1^2+3^2)=sqrt(10)}}}

the magnitude of {{{v }}}is {{{sqrt((-3)^2+1^2)=sqrt(10)}}}


{{{cos(theta) = 0/(sqrt(10)*sqrt(10))}}}

{{{cos(theta) = 0}}}

{{{theta = cos^-1(0)}}}

{{{theta = 90}}}°