Question 1206882

{{{u = i + 6j}}}, {{{v = 7i - 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]=6}}}
{{{a[2]=7}}}, {{{b[2]=-1}}}


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

{{{u *v = (1 *7) + (6 *-1) =7-6=1}}}



(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+6^2)=sqrt(37)}}}

the magnitude of {{{v }}}is {{{sqrt(7^2+(-1)^2)=sqrt(50)}}}


{{{cos(theta) = 1/(sqrt(37)*sqrt(50))}}}

{{{cos(theta) = 0.02324952774876386}}}

{{{theta = cos^-1(0.02324952774876386)}}}

{{{theta = 88.667780146130361}}}°

{{{theta = 89}}}°