Find the volume of the tetrahedron with the vertices $P(1,1,1)$, $Q(1, 2, 3)$, $R(3, 1, 2)$, and $S(2, 3, 1)$.
Finding the volume of the tetrahedron.
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geometry
volume
1 Answers
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Volume of a tetrahedron is $$\dfrac13 \times \text{Base area} \times \text{height}$$ If the vertices are $\vec{v_1},\vec{v_2},\vec{v_3},\vec{v_4}$, then the volume is given by $$\left \vert \dfrac{(\vec{v}_2 - \vec{v}_1) \cdot \left((\vec{v}_3 - \vec{v}_1) \times (\vec{v}_4 - \vec{v}_1) \right)}6 \right \vert$$ where $\vec{a} \cdot \vec{b}$ denotes the inner/dot product, and $\vec{a} \times \vec{b}$ denotes the cross product.
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0so PQ QR RS PS will be my 4 vertices, right? – 2012-10-30
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0@Yigit No. P, Q ,R and S are your vertices. – 2012-10-30
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0Oh i see, but here in your solution, the order of given vertices is important. What would happen if it is given in different order? The v1 v2 vertices will become different. Does this change the result? – 2012-10-30
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0@Yigit: Good observation. But the interesting thing is that the result is invariant (this is a good exercise to prove it is invariant, though it is not that hard. Just expand the dot product and cross product) except probably for the signs. That is why I have a $\vert \cdot \vert$ to convert it into a positive quantity always. – 2012-10-30
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0Oh the magic of Math. interesting, thank you. – 2012-10-30