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Scalar Triple Product
Scalar triple product is the dot product of a vector with the cross product of two other vectors, i.e., if a, b, c are three vectors, then their scalar triple product is a · (b × c). It is also commonly known as the triple scalar product, box product, and mixed product. The scalar triple product gives the volume of a parallelepiped, where the three vectors represent the adjacent sides of the parallelepiped.
In this article, we will explore the concept of the scalar triple product, its formula, proof, and properties. We will also study the geometric interpretation of the scalar triple product and solve a few examples based on the concept to understand its application.
1.  What is Scalar Triple Product? 
2.  Scalar Triple Product Formula 
3.  Geometrical Interpretation of Scalar Triple Product 
4.  Properties of Scalar Triple Product 
5.  FAQs on Scalar Triple Product 
What is Scalar Triple Product?
The scalar triple product of three vectors a, b, c is the scalar product of vector a with the cross product of the vectors b and c, i.e., a · (b × c). Symbolically, it is also written as [a b c] = [a, b, c] = a · (b × c). The scalar triple product [a b c] gives the volume of a parallelepiped with adjacent sides a, b, and c. If we are given three vectors a, b, c, then their scalar triple products [a b c] are:
 a · (b × c)
 a · (c × b)
 b · (a × c)
 b · (c × a)
 c · (b × a)
 c · (a × b)
Now, before moving to the formula of the scalar triple product, we need to note that:
 [a, b, c] = a · (b × c) = b · (c × a) = c · (a × b)
 a · (b × c) =  a · (c × b)
 b · (c × a) =  b · (a × c)
 c · (a × b) =  c · (b × a)
 a · (b × c) = (a × b) · c
Scalar Triple Product Formula
If we are given three vectors a = a_{1} i + a_{2 }j + a_{3} k, b = b_{1} i + b_{2 }j + b_{3} k, and c = c_{1} i + c_{2 }j + c_{3} k, then their scalar triple product is given by the determinant of the components of the three vectors. The formula for the scalar triple product of vectors a, b, c is given by,
Scalar Triple Product Proof
Now, we will prove the formula for the scalar triple product of three vectors a, b, c. Using the definition of the cross product and dot product, we have
a · (b × c) = \(\overrightarrow{a} \cdot \left\begin{array}{lll}\hat{i} & \hat{j} & \hat{k} \\b_{1} & b_{2} & b_{3} \\c_{1} & c_{2} & c_{3}\end{array}\right\)
= \([(b_2c_3  c_2b_3)\hat{i}  (b_1c_3c_1b_3)\hat{j} + (b_1c_2c_1b_2)\hat{k}] \cdot (a_1\hat{i} + a_2\hat{j} + a_3\hat{k})\)
= (b_{2}c_{3}  c_{2}b_{3})a_{1} + (c_{1}b_{3}  b_{1}c_{3})a_{2} + (b_{1}c_{2}  c_{1}b_{2})a_{3}
= \( \left\begin{array}{lll}a_1 & a_2& a_3 \\b_{1} & b_{2} & b_{3} \\c_{1} & c_{2} & c_{3}\end{array}\right\)
Hence, we have proved the formula of scalar triple product of three vectors a, b, c.
Geometrical Interpretation of Scalar Triple Product
Now, we know that given any three vectors a, b, c, the scalar triple product is given by a · (b × c) which is equal to the determinant of the components of the three vectors. Let us now understand the geometrical interpretation of the scalar triple product. The absolute value of the scalar triple product a · (b × c) gives the volume of a parallelepiped, where a, b, c form the adjacent sides of the parallelopiped. The cross product (b × c) gives the area of the parallelogram formed by the vectors b and c. Using the definition cross product, b × c is perpendicular to the plane containing vectors b and c.
Properties of Scalar Triple Product
We have explored the concept of the scalar triple product along with its geometrical interpretation and formula. Let us now go through some of its important properties for a better understanding of the concept:
 The scalar triple product of three vectors is zero if any two of them are parallel, i.e., [a a b] = 0
 [(a + b) c d] = [a c d] + [b c d]
LHS = [(a + b) c d]
= (a + b) · (c × d)
= a · (c × d) + b · (c × d)
= [a c d] + [b c d]
= RHS  [λa b c] = λ [a b c], where λ is a real number.
 The scalar triple product of three nonzero vectors is zero if and only if they are coplanar.
 Since the scalar product is commutative, therefore we have
 a · (b × c) = (b × c) · a
 b · (c × a) = (c × a) · b
 c · (a × b) = (a × b) · c
Important Notes on Scalar Triple Product
 [a, b, c] = [b, c, a] = [c, a, b]
 [a (b+c) d] = [a b d] + [a c d], [a b (c+d)] = [a b c] + [a b d]
 [λa b c] = [a λb c] = [a b λc] = λ [a b c], where λ is a real number.
 The scalar triple product of three nonzero vectors is zero if and only if they are coplanar.
Related Topics on Scalar Triple Product
Scalar Triple Product Examples

Example 1: Evaluate the volume of a parallelepiped whose coterminous edges are i  j + k, 2i + 3j  k, and i  j + 5k.
Solution: To determine the volume of the parallelepiped with edges i  j + k, 2i + 3j  k, and i  j + 5k, we will determine its scalar triple product.
[ i  j + k, 2i + 3j  k, i  j + 5k] = \(\left\begin{array}{lll}1 & 1 & 1 \\2 & 3 & 1 \\1 & 1 & 5\end{array}\right\)
= 1(15  1) + 1(10  1) + 1(2 + 3)
= 14 + 9 + 1
= 24
Answer: The volume of the parallelepiped is 24 cubic units.

Example 2: Show that the vectors i + 2j − 3k, 2i − j + 2k and 3i + j − k are coplanar.
Solution: We know that the scalar triple product of three nonzero vectors is zero if and only if they are coplanar. So, we will show that the scalar triple product of the given vectors is equal to zero.
[ i + 2j − 3k, 2i − j + 2k, 3i + j − k] = \(\left\begin{array}{lll}1 & 2 & 3 \\2 & 1 & 2 \\3 & 1 & 1\end{array}\right\)
= 1(1  2)  2(2  6)  3(2 + 3)
= 1 + 16  15
= 0
Answer: Hence, we proved that the given vectors are coplanar.
FAQs on Scalar Triple Product
What is Scalar Triple Product in Vector Theory?
Scalar triple product is the dot product of a vector with the cross product of two other vectors, i.e., if a, b, c are three vectors, then their scalar triple product is a · (b × c).
What is Scalar Triple Product Formula?
The formula for the scalar triple product of vectors a = a_{1} i + a_{2 }j + a_{3} k, b = b_{1} i + b_{2 }j + b_{3} k, and c = c_{1} i + c_{2 }j + c_{3} k is given by, \( \left\begin{array}{lll}a_1 & a_2& a_3 \\b_{1} & b_{2} & b_{3} \\c_{1} & c_{2} & c_{3}\end{array}\right\)
Why is the Scalar Triple Product of Three Coplanar Vectors Zero?
Assume a, b, c are three nonzero vectors. Then [a, b, c] = 0 ⇔ (a × b) · c = 0 ⇔ c is perpendicular to a × b ⇔ c lies in the plane parallel to both a and b ⇔ a, b, c are coplanar.
When is Scalar Triple Product Zero?
The scalar triple product of three vectors is zero if any two of them are equal to parallel vectors.
What is the Geometrical Interpretation of the Scalar Triple Product of Three Vectors?
The geometrical interpretation of the scalar triple product of three vectors is that it gives the volume of a parallelepiped and the three vectors represent the coterminous edges of the parallelepiped. If the scalar triple product is zero, then the volume will be zero and implies that all edges lie on the same plane, and hence the vectors are coplanar.
Why is Scalar Triple Product called the Box Product?
The scalar triple product of three vectors a, b, c is written in a box as [a, b, c]. Also, the absolute value of the scalar triple product gives the volume of a box (parallelepiped).
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