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Geometry

Dot Product of Vectors

Dot product - Area Calculations

         

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

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

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

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

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

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