Maths · JEE Advanced only

Scalar and Vector Triple Products (JEE Advanced)

Not in the JEE Main syllabus, but it is in JEE Advanced. Study time: about 4-6 hours to learn concepts and formulas, practice determinant calculations, and solve variety of problems involving volumes and coplanarity.

Last updated: 26 October 2026

Direct answer

Scalar triple product gives you the volume of a parallelepiped formed by three vectors, while vector triple product gives you a vector that lies in the plane of two of the original vectors.

Scalar triple product gives you the volume of a parallelepiped formed by three vectors, while vector triple product gives you a vector that lies in the plane of two of the original vectors. These operations extend your toolkit beyond simple dot and cross products to solve problems involving volumes, coplanarity, and complex vector relationships.

Official syllabus

What the JEE Advanced syllabus says

“scalar and vector triple products, and their geometrical interpretations”

Source: JEE Advanced 2026 Information Bulletin, Annexure-I. Closest JEE Main topic: scalar and vector products.

JEE Main only tests scalar and vector products of two vectors, so you need to learn how three vectors interact through these triple products. Advanced loves asking about volumes, coplanarity conditions, and vector identities that require these concepts.

Key ideas

What you must understand

Scalar triple product [a b c] = a · (b × c) represents the signed volume of the parallelepiped with edges a, b, c
Three vectors are coplanar if and only if their scalar triple product equals zero
Scalar triple product is unchanged by cyclic permutation: [a b c] = [b c a] = [c a b]
Vector triple product follows the BAC-CAB rule: a × (b × c) = b(a · c) − c(a · b)
The result of a × (b × c) always lies in the plane containing b and c
Scalar triple product can be computed as a 3×3 determinant of the component vectors

Formulas

Formulas and key facts

[a b c] = a · (b × c) = |a₁ a₂ a₃; b₁ b₂ b₃; c₁ c₂ c₃| (determinant)
Volume of parallelepiped = |[a b c]|
Volume of tetrahedron = (1/6)|[a b c]|
a × (b × c) = b(a · c) − c(a · b) (BAC-CAB rule)
(a × b) × c = b(a · c) − a(b · c)
[a b c] = [b c a] = [c a b] = −[b a c] = −[a c b] = −[c b a]
Coplanarity condition: [a b c] = 0
[a + b, c, d] = [a c d] + [b c d] (linearity)

Avoid

Common mistakes

Confusing a × (b × c) with (a × b) × c, they are NOT equal and have different formulas
Forgetting the negative sign when swapping two vectors in scalar triple product
Using the wrong formula: a × (b × c) = b(a · c) − c(a · b), not b(a · b) − c(a · c)
Forgetting the factor of 1/6 when finding tetrahedron volume from scalar triple product

Worked example

A JEE Advanced-level example

Question. If a = i + j + k, b = i + 2j + 3k, and c = 2i + 3j + k, find the volume of the tetrahedron with vertices at origin and points A, B, C with position vectors a, b, c.

Show solution

Volume of tetrahedron = (1/6)|[a b c]| = (1/6)|a · (b × c)|
First find b × c: b × c = |i j k; 1 2 3; 2 3 1| = i(2−9) − j(1−6) + k(3−4) = −7i + 5j − k
Now find a · (b × c): a · (b × c) = (1)(−7) + (1)(5) + (1)(−1) = −7 + 5 − 1 = −3
So [a b c] = −3, and |[a b c]| = 3
Volume of tetrahedron = (1/6) × 3 = 1/2 cubic units

Answer: 1/2 cubic units

Solved twice independently; both solutions agree.

FAQs

Frequently asked questions

Is Scalar and Vector Triple Products in the JEE Main syllabus?

No. It is in the JEE Advanced 2026 syllabus only. The closest JEE Main topic is scalar and vector products.

How long does Scalar and Vector Triple Products take to prepare for JEE Advanced?

About 4-6 hours to learn concepts and formulas, practice determinant calculations, and solve variety of problems involving volumes and coplanarity, if you already know the related JEE Main topic.