Scalar Triple Product

The scalar triple product combines both the dot and cross products into a single operation involving three vectors. Its result is a scalar quantity equal to the volume of the parallelepiped formed by the three vectors. This operation appears in EM in coordinate system transformations, in verifying coplanarity of vectors, and in deriving volume elements in curvilinear systems. It is also a standard GATE topic tested through determinant-based problems.

Core Concept Explanation

The scalar triple product is written as A dot (B cross C), often denoted compactly as [A B C]. The operation proceeds in two steps: first compute B cross C (a vector result), then take the dot product of A with that result (producing a scalar). This scalar equals the signed volume of the parallelepiped formed by using A, B, and C as three adjacent edge vectors. The sign indicates orientation: positive if A, B, C form a right-handed system, negative if left-handed.

The most important property for computation is the determinant form. The scalar triple product equals the value of the 3 by 3 determinant whose rows are the Cartesian components of A, B, and C in that order. This makes calculation fast: set up the determinant and expand along the first row. This determinant-based approach is universally preferred in GATE problems.

An extremely important geometric consequence is coplanarity: if [A B C] = 0, then the three vectors are coplanar (they all lie in the same plane). This means the parallelepiped has collapsed into a flat sheet with zero volume. This condition is used in problems asking to verify whether three given vectors lie in a common plane, or to find a constant k such that a given set of vectors becomes coplanar.

Mathematical Expression

The determinant expansion of [A B C] along the first row is: Ax(ByBz minus BzCy) minus Ay(BxCz minus BzCx) plus Az(BxCy minus ByCx). Rather than memorizing this expansion, it is better to set up the 3 by 3 determinant each time and use the standard cofactor expansion. This avoids sign errors.

The cyclic permutation property states that A dot (B cross C) = B dot (C cross A) = C dot (A cross B). All three cyclic permutations give the same scalar value. However, swapping any two adjacent vectors negates the result: A dot (B cross C) = negative A dot (C cross B). This anti-symmetry under swap follows from the anti-commutativity of the cross product.

Practical Understanding

In electromagnetics, scalar triple products arise when computing volume integrals in oblique coordinate systems. The Jacobian of a coordinate transformation involves a triple product of the basis vectors, and its value determines how volumes scale between the two coordinate systems. When working in rectangular coordinates, the triple product of unit vectors ax dot (ay cross az) = ax dot az... wait, ay cross az = ax, and ax dot ax = 1, confirming that the unit volume in Cartesian coordinates is 1.

In circuit analysis related contexts, the scalar triple product helps verify whether a set of loop vectors in network analysis spans a three-dimensional space or are linearly dependent (coplanar). Linear dependence of three vectors is exactly the condition [A B C] = 0, which corresponds to a singular system with no unique solution.

Given:
A = 1ax + 2ay + 3az
B = 0ax + 1ay + 1az
C = 2ax + 0ay + 1az

Why this formula applies:
Three vectors given in Cartesian form; use 3×3 determinant directly.

Formula:
[A B C] = A · (B × C) = determinant of 3×3 matrix

Substitution:
| 1   2   3 |
| 0   1   1 |
| 2   0   1 |

Calculation (expand along row 1):
= 1×(1×1 - 1×0) - 2×(0×1 - 1×2) + 3×(0×0 - 1×2)
= 1×(1 - 0) - 2×(0 - 2) + 3×(0 - 2)
= 1×1 - 2×(-2) + 3×(-2)
= 1 + 4 - 6
= -1

Final Answer:
[A B C] = -1 (scalar)
Since result ≠ 0, vectors are NOT coplanar.
Signed volume of parallelepiped = |-1| = 1 cubic unit

Exam Tip: If GATE asks 'for what value of k are vectors A, B, C coplanar?', set the determinant [A B C] = 0 and solve for k. This is a one-equation one-unknown problem. Also remember: cyclic permutation preserves the value, swapping two vectors negates it.

Key Points on Scalar Triple Product Mechanism

• A dot (B cross C) = 3x3 determinant with rows A, B, C. This is the fastest computational route.
• Result is a scalar equal to the signed volume of the parallelepiped formed by A, B, C.
• If [A B C] = 0, the three vectors are coplanar and linearly dependent.
• Cyclic permutations preserve the value: [A B C] = [B C A] = [C A B].
• Swapping any two adjacent vectors negates the result: [A B C] = -[B A C].
• The dot and cross positions can be exchanged: A dot (B cross C) = (A cross B) dot C.

Quick Revision

• [A B C] = A · (B × C) = 3x3 determinant. Memorize this as the definition and computation method together.
• Result is a scalar representing volume of parallelepiped. |[A B C]| is the unsigned volume.
• Zero value means coplanar. GATE uses this to find unknown k in a vector set.
• Cyclic permutation: [A B C] = [B C A] = [C A B] — all equal.
• Anti-cyclic: swapping any two gives negative: [A C B] = -[A B C].
• GATE trap: the scalar triple product is a NUMBER, not a vector. If your answer has ax or ay, something went wrong.
• Dot and cross are interchangeable: A·(B×C) = (A×B)·C. Both give the same scalar.