Question:** A tetrahedron has vertices at \((0,0,0)\), \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\). What is the volume of this tetrahedron?
["Understanding the Volume of a Tetrahedron with Given Vertices", "When calculating the volume of a 3D tetrahedron defined by four points in space, understanding the geometry of the shape is essential. One classic example involves a tetrahedron with vertices at ((0,0,0)), ((1,0,0)), ((0,1,0)), and ((0,0,1)). This simple yet insightful configuration provides a clear pathway to determining its volume using basic mathematics and vector geometry.", "### The Geometry Behind the Tetrahedron", "The four vertices ((0,0,0)), ((1,0,0)), ((0,1,0)), and ((0,0,1)) form a tetrahedron rooted at the origin and extending one unit along each of the (x), (y), and (z) axes. These three edges along the coordinate axes create a right-angled tetrahedron with mutually perpendicular edges meeting at the origin. This orthogonality simplifies the volume calculation significantly.", "### Formula for Volume Using Vectors", "The volume ( V ) of a tetrahedron defined by four points ( A, B, C, D ) can be computed using the scalar triple product of vectors:", "[\nV = \frac{1}{6} \left| \vec{AB} \cdot (\vec{AC} \ imes \vec{AD}) \right|\n]", "Where:\n- ( \vec{AB}, \vec{AC}, \vec{AD} ) are vectors from point ( A ) to ( B ), ( C ), and ( D ) respectively.\n- The cross product ( \vec{AC} \ imes \vec{AD} ) gives a vector perpendicular to the plane formed by ( AC ) and ( AD ), and the dot product with ( \vec{AB} ) yields the signed volume.", "For our specific tetrahedron:\n- Let ( A = (0,0,0) ), ( B = (1,0,0) ), ( C = (0,1,0) ), ( D = (0,0,1) )\n- Then:\n [\n \vec{AB} = \langle 1, 0, 0 \rangle, \quad\n \vec{AC} = \langle 0, 1, 0 \rangle, \quad\n \vec{AD} = \langle 0, 0, 1 \rangle\n ]", "### Step-by-Step Computation", "First, compute the cross product ( \vec{AC} \ imes \vec{AD} ):", "[\n\vec{AC} \ imes \vec{AD} = \n\begin{vmatrix}\n\mathbf{i} & \mathbf{j} & \mathbf{k} \\n0 & 1 & 0 \\n0 & 0 & 1 \\n\end{vmatrix}\n= \mathbf{i}(1 \cdot 1 - 0 \cdot 0) - \mathbf{j}(0 \cdot 1 - 0 \cdot 0) + \mathbf{k}(0 \cdot 0 - 0 \cdot 1) = \langle 1, 0, 0 \rangle\n]", "Now compute the dot product ( \vec{AB} \cdot (\vec{AC} \ imes \vec{AD}) ):", "[\n\vec{AB} \cdot \langle 1, 0, 0 \rangle = (1, 0, 0) \cdot (1, 0, 0) = 1\n]", "Thus, the volume is:", "[\nV = \frac{1}{6} \left| 1 \right| = \frac{1}{6}\n]", "### Why This Volume Equals ( \frac{1}{6} )", "Geometrically, this tetrahedron is one of the six identical tetrahedra into which the unit cube ([0,1]^3) can be divided by connecting the origin to the faces. Each tetrahedron occupies exactly ( \frac{1}{6} ) of the cube’s volume (which is 1), confirming the computed result.", "### Conclusion", "The volume of the tetrahedron with vertices ((0,0,0)), ((1,0,0)), ((0,1,0)), and ((0,0,1)) is exactly ( \frac{1}{6} ) cubic units. This straightforward example highlights how symmetry, orthogonality of edges, and vector calculus simplify volume computation in analytic geometry. Mastering such calculations is invaluable in fields ranging from computer graphics to geometric optimization in architecture and engineering.", "If you’re studying spatial reasoning or preparing for advanced math or physics, understanding tetrahedral geometry enhances your ability to visualize and compute three-dimensional volumes—critical skills in both theoretical and applied sciences."]
