Question

Recall: Every vector ( vec{a}=leftlangle a_{1}, a_{2}, a_{3} ight angle ) in ( mathbb{R}^{3} ) can be represented as ( a_{1} vec{i}+a_{2} vec{j}+a_{3} vec{k} ), where [ vec{i}=langle 1,0,0 angle quad vec{j}=langle 0,1,0 angle quad vec{k}=<0,0,1 angle ] are the standard basis vectors in the direction of the positive ( x-, y- ), and ( z-a x e s ). The vectors ( a_{1} overrightarrow{oldsymbol{i}}, a_{2} overrightarrow{oldsymbol{j}} ), and ( a_{3} overrightarrow{oldsymbol{k}} ) are said to be the components of the vector ( overrightarrow{oldsymbol{a}} ) since they add together to yield ( overrightarrow{oldsymbol{a}} ). Given points ( P(-3,2,0), Q(5,-1,1) ), and ( R(1,3,-2) ). 1. Find the vectors ( overrightarrow{oldsymbol{P Q}}, overrightarrow{P R} ), and ( overrightarrow{Q R} ). Express your answer in component form. 2. Find a unit vector that is perpendicular to vectors ( overrightarrow{oldsymbol{P Q}} ) and ( overrightarrow{oldsymbol{P R}} ). 3. Find the area of the triangle ( P Q R ).

          Recall: Every vector ( vec{a}=leftlangle a_{1}, a_{2}, a_{3}
ight
angle ) in ( mathbb{R}^{3} ) can be represented as ( a_{1} vec{i}+a_{2} vec{j}+a_{3} vec{k} ), where
[
vec{i}=langle 1,0,0
angle quad vec{j}=langle 0,1,0
angle quad vec{k}=<0,0,1
angle
]
are the standard basis vectors in the direction of the positive ( x-, y- ), and ( z-a x e s ). The vectors ( a_{1} overrightarrow{oldsymbol{i}}, a_{2} overrightarrow{oldsymbol{j}} ), and ( a_{3} overrightarrow{oldsymbol{k}} ) are said to be the components of the vector ( overrightarrow{oldsymbol{a}} ) since they add together to yield ( overrightarrow{oldsymbol{a}} ).

Given points ( P(-3,2,0), Q(5,-1,1) ), and ( R(1,3,-2) ).
1. Find the vectors ( overrightarrow{oldsymbol{P Q}}, overrightarrow{P R} ), and ( overrightarrow{Q R} ). Express your answer in component form.
2. Find a unit vector that is perpendicular to vectors ( overrightarrow{oldsymbol{P Q}} ) and ( overrightarrow{oldsymbol{P R}} ).
3. Find the area of the triangle ( P Q R ).

Recall: Every vector ( veca=leftlangle a1, a2, a3
ight
angle ) in ( mathbbR^3 ) can be represented as ( a1 veci+a2 vecj+a3 veck ), where
[
veci=langle 1,0,0
angle quad vecj=langle 0,1,0
angle quad veck=<0,0,1
angle
]
are the standard basis vectors in the direction of the positive ( x-, y- ), and ( z-a x e s ). The vectors ( a1 overrightarrowoldsymboli, a2 overrightarrowoldsymbolj ), and ( a3 overrightarrowoldsymbolk ) are said to be the components of the vector ( overrightarrowoldsymbola ) since they add together to yield ( overrightarrowoldsymbola ).

Given points ( P(-3,2,0), Q(5,-1,1) ), and ( R(1,3,-2) ).
1. Find the vectors ( overrightarrowoldsymbolP Q, overrightarrowP R ), and ( overrightarrowQ R ). Express your answer in component form.
2. Find a unit vector that is perpendicular to vectors ( overrightarrowoldsymbolP Q ) and ( overrightarrowoldsymbolP R ).
3. Find the area of the triangle ( P Q R ).

Added by Joel W.

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