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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.

# 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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