Volume Of Parallelepiped

Question

Three coterminous edges of a parallelepiped are a = i + j, b = j + k and c = k + i; calculate the volume of this solid.

RankGuru · Accepted Answer

The volume of a parallelepiped with coterminous edges a, b, c equals the absolute value of the scalar triple product |[a b c]|, evaluated as the magnitude of the determinant of their components. This identity is a staple of JEE Advanced solid-geometry-via-vectors questions. Arrange the components into rows (1,1,0), (0,1,1), (1,0,1). Expand along the first row: 1[(1)(1) - (1)(0)] - 1[(0)(1) - (1)(1)] + 0[(0)(0) - (1)(1)] = 1(1 - 0) - 1(0 - 1) + 0 = 1 - 1(-1) = 1 + 1 = 2. The volume is therefore |2| = 2 cubic units. Option 1 results from dropping the second cofactor's sign contribution. Option 3 over-counts by mishandling the zero term. Option 4 doubles the determinant erroneously. The nonzero volume confirms the three edge vectors are linearly independent and truly span a three-dimensional solid. Plausibility check: each edge has length sqrt(2), and a roughly cubic arrangement of such edges should give a volume of order a few units, so 2 is dimensionally and numerically sensible.

Volume Of Parallelepiped

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