Finding the volume of the tetrahedron.

Question

Find the volume of the tetrahedron with the vertices $P(1,1,1)$, $Q(1, 2, 3)$, $R(3, 1, 2)$, and $S(2, 3, 1)$.

Answer 1

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.