Question 364466
b) Use the dot product.
{{{a}}}=({{{3}}},{{{1}}},{{{-2}}})
{{{b}}}=({{{-6}}},{{{-2}}},{{{4}}})
{{{a*b}}}={{{3(-6)+1(-2)+(-2)(4)=-18-2-8=-28}}}
{{{a*b=abs(a)abs(b)cos(theta)}}}
{{{abs(a)=sqrt(3^2+1^2+(-2)^2)=sqrt(9+1+4)=sqrt(14)}}}
{{{abs(b)=sqrt((-6)^2+(-2)^2+(4)^2)=sqrt(36+4+16)=sqrt(56)=2sqrt(14)}}}
{{{-28=sqrt(14)(2*sqrt(14))cos(theta)}}}
{{{cos(theta)=-28/(2*14)}}}
{{{cos(theta)=-1}}}
{{{highlight(theta=180)}}} 
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c) 
Perpendicular means {{{theta=90}}}.
{{{cos(90)=0}}}
So then {{{a*b=0}}}
{{{a}}}=({{{5}}},{{{2}}},{{{3}}})
{{{b}}}=({{{2}}},{{{alpha}}},{{{alpha}}})
{{{a*b}}}={{{5*2+2*alpha+3*alpha=10+5*alpha}}}
{{{10+5*alpha=0}}}
{{{5*alpha=-10}}}
{{{highlight(alpha=-2)}}}