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Dot Product of Vectors

Dot product - Area Calculations


Consider the three points A=(2,3,4),B=(1,5,3),A=(2,3,4),B=(1,5,3), and C=(2,7,4).C=(2,7,4). What is the area of ABC?\triangle ABC?

Consider a triangle OAB.\triangle OAB. If OA=(0,2,2)\overrightarrow{OA}=(0,2,-2) and OB=(1,4,1),\overrightarrow{OB}=(1,-4,1), then what is the area of OAB?\triangle OAB?

If A=(0,1,1),B=(1,2,3),A=(0,1,1),B=(-1,2,3), and C=(4,0,1),C=(4,0,-1), what is the area of ABC?\triangle ABC?

Consider a parallelogram ABCD.ABCD. If A=(2,3,1),B=(0,5,3),A=(-2,3,1),B=(0,5,3), and D=(1,7,4),D=(1,7,4), what is the square of the area of ABCD?ABCD?

Calculate the area of the triangle spanned by the vectors a \vec{a} and b, \vec{b}, where a=3, \lvert \vec{a} \rvert = 3 , b=4 \lvert \vec{b} \rvert = 4 and ab=82. \vec{a} \cdot \vec{b} = 8\sqrt{2}.


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