Vectors: Scalar & Vector Triple Products JEE Main Guide

The scalar triple product and vector triple product are the most advanced vector concepts tested in JEE Main Mathematics and Physics. In Math, they appear directly in 3D Geometry questions (volume of parallelepiped, coplanarity of vectors). In Physics, they appear in cross-product force and torque calculations. This guide covers both products systematically with the exact JEE question patterns.

Scalar Triple Product (Box Product)

The scalar triple product of vectors a, b, c is [a b c] = a·(b×c) = b·(c×a) = c·(a×b). In Cartesian form: [a b c] = det([a₁ a₂ a₃; b₁ b₂ b₃; c₁ c₂ c₃]). Key properties: (1) Cyclic permutations do not change the value: [a b c] = [b c a] = [c a b]; (2) Interchange of any two vectors changes the sign: [a b c] = −[b a c]; (3) [a b c] = 0 iff a, b, c are coplanar (linearly dependent); (4) |[a b c]| = volume of the parallelepiped with edges a, b, c. If the edges of a parallelepiped are 3a, 4b, 2c and [a b c] = V, the volume of the parallelepiped = |[3a 4b 2c]| = 3·4·2·|[a b c]| = 24|V|.

Coplanarity condition for four points A, B, C, D: vectors AB, AC, AD are coplanar, i.e., [AB AC AD] = 0. JEE asks: "Show that the points A, B, C, D are coplanar" or "For what value of λ are the four points coplanar?" — always compute the scalar triple product. Test your vector product skills with a free mock. For the complete vector foundations, see our vector algebra guide.

Applications: Area, Volume, and Projection

Area of triangle with vertices A, B, C: Area = ½|AB×AC|. Area of parallelogram spanned by vectors a and b: |a×b|. Volume of parallelepiped with edges a, b, c: |[a b c]|. Volume of tetrahedron with edges a, b, c from one vertex: (1/6)|[a b c]|. JEE uses these area and volume formulae in both direct calculation questions and in "find the value of λ such that the volume is..." questions.

Projection of vector a on vector b: (a·b)/|b| = |a|cosθ where θ is the angle between a and b. Vector projection of a onto b: (a·b)/|b|² · b. Projection of a onto the plane perpendicular to b: a − (a·b)/|b|² · b. These projection formulae appear in Physics (work done = F·d projected on displacement direction) and in Math (3D Geometry problems involving shortest distances and angles). For the cross product and torque applications in Physics, see our torque and angular momentum guide.

Vector Triple Product

The vector triple product a×(b×c) = (a·c)b − (a·b)c. Mnemonic: "BAC minus CAB" — the result is the first outer vector times the dot product of the outer with the last, minus the last outer times the dot product of the outer with the middle: a×(b×c) = (a·c)b − (a·b)c. Note: (a×b)×c = (a·c)b − (b·c)a — a different expression (the product is not associative). JEE occasionally asks to simplify a vector triple product expression — always apply this identity directly.

Key identities involving scalar and vector triple products: a·(b×c) = (a×b)·c (the dot and cross can be swapped). [a+b, b+c, c+a] = 2[a b c] (the box product of "shifted" vectors equals twice the original). [ka, lb, mc] = klm[a b c]. These identities are tested in "find the value of [a+b b+c c+a]" type questions where the answer is always 2[a b c]. For related 3D applications, see our vectors dot and cross products guide.

Exam Strategy for Vector Questions

Vector questions in JEE Main follow three main formats: (1) Pure computation — given the components, compute a dot/cross/triple product (fastest questions, should take 60–90 seconds); (2) Condition problems — find λ such that vectors satisfy some property (coplanar, perpendicular, etc.) — set up the condition, solve for λ; (3) Geometric application — find area, volume, projection, or angle (set up the correct formula from the problem description). The common error in format 3 is setting up the wrong formula — always verify what formula applies before computing. For the complete 3D Geometry and Vectors preparation, see our 3D geometry and vectors guide and our 3D distance and angle guide.