Vectors: Advanced (Triple Product, Coplanarity)

Advanced vector algebra centers on triple products -- scalar and vector -- and their geometric applications including coplanarity, volume computation, and linearity tests. These tools extend the dot and cross product framework to three-vector interactions and are essential for 3D geometry problem-solving.

Scalar Triple Product (STP): For vectors a, b, c, the scalar triple product [a b c] = a.(b x c) = the determinant of the 3x3 matrix formed by the components of a, b, c as rows (or columns). Geometrically, |[a b c]| equals the volume of the parallelepiped formed by a, b, c. The sign indicates orientation: positive if a, b, c form a right-handed system.

Properties of STP: [a b c] = [b c a] = [c a b] (cyclic permutation preserves value). Swapping any two vectors negates the sign: [a b c] = -[b a c]. If any two vectors are equal or proportional, [a b c] = 0. The STP is linear in each argument.

Coplanarity: Three vectors a, b, c are coplanar if and only if [a b c] = 0. Four points A, B, C, D are coplanar iff [AB AC AD] = 0. This is the primary test for coplanarity in JEE problems.

Vector Triple Product (VTP): a x (b x c) = (a.c)b - (a.b)c (BAC-CAB rule). This always lies in the plane of b and c. Note: (a x b) x c = (a.c)b - (b.c)a, which lies in the plane of a and b. The cross product is NOT associative: a x (b x c) != (a x b) x c in general.

Reciprocal System of Vectors: Given non-coplanar vectors a, b, c, the reciprocal vectors are a' = (b x c)/[a b c], b' = (c x a)/[a b c], c' = (a x b)/[a b c]. These satisfy a.a' = b.b' = c.c' = 1 and a.b' = a.c' = 0, etc. The reciprocal system is used in crystallography and resolving vectors along non-orthogonal directions.

Volume and Area via Products: Volume of tetrahedron with vertices A, B, C, D = ($\frac{1}{6}$)|[AB AC AD]|. Volume of parallelepiped = |[a b c]|. Area of triangle with sides a, b = ($\frac{1}{2}$)|a x b|.

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