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Dot Product of Vectors

The dot product (also known as the scalar product) is an operation on vectors that can tell you the angle between the vectors.

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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