103 LearnersLast updated on July 10th, 2025
Scalar Triple Product
The scalar triple product is the dot product of a vector with the cross product of two other vectors. It is represented as a · (b × c), where a, b, c are vectors. It is also known as the box product or scalar triple product.
What is a Scalar Triple Product?
The mathematical operation involving a vector operation in a three-dimensional space is the scalar triple product. It involves the dot product of one vector with the cross product of two others. The result is a scalar value. Mathematically, the scalar triple product of the vectors a, b, and c can be represented as:
a · (b × c)
Scalar Triple Product vs Vector Triple Product
The two vector operations are the scalar triple product and vector triple product are two distinct vector operations. In this section, we will learn the difference between the scalar triple product and the vector triple product.
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Scalar Triple Product |
Vector Triple Product |
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For vectors a, b, and c, the scalar triple product is represented as a · (b × c) |
For vectors a, b, and c, the vector triple product is represented as a × (b × c) |
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The result will be a scalar |
The result will be a vector |
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In geometric interpretation, the scalar triple product represents the volume of the parallelepiped formed by vectors a, b, and c. |
The geometric interpretation of the vector triple product gives the directed volume of a parallelepiped |
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It is antisymmetric as the sign changes with the order of the vectors |
The vector triple product is not antisymmetric; changing the order of vectors alters the resulting vector. |
Properties of Scalar Triple Product
In this section, we will learn the properties of scalar triple products. Understanding these properties helps students calculate volumes and analyze vector orientation and relationships in three-dimensional spaces.
- Antisymmetric: Exchanging any two vectors in the scalar triple product changes its sign.
- For any three vectors A, B, and C
A · (B × C) = B · (C × A) = C · (A × B)
-A · (C × B) = -B · (A × C) = -C · (B × A)
- Linearity: With respect to each vector, the scalar triple product is linear
[A+D,B,C] = [A,B,C] + [D,B,C]
- Cyclic Permutation: If the vectors are cyclically permuted, then the scalar triple product remains unchanged
A · (B × C) = B · (C × A) = C · (A × B)
- Volume interpretation: The signed volume of the parallelepiped formed by vectors A, B, and C is represented by the scalar triple product. The absolute value represents the actual volume, and the sign indicates the orientation of the three vectors.
- Orthogonality: The scalar triple product is zero if the vectors are coplanar.
What is the Formula for Scalar Triple Product?
For any three vectors, a, b, and c, the scalar triple product is represented as a · (b × c). The result of this operation is scalar, and it can be calculated using the determinant of a 3 × 3 matrix formed by the components of the vectors:
The vectors in component form:
a = a1 i +a2 j +a3 k
b = b1 i +b2 j +b3 k
c = c1 i +c2 j +c3 k
Finding the scalar triple product a · (b × c) is:
Finding the cross product of b × c
b × c is calculated using the determinant:
(b2c3 - c2b3) i - (b1c3 - c1b3) j + (b1c2 - c1b2) k
Finding the dot product with vector a:
[(b2c3 - c2b3) i - (b1c3 - c1b3) j + (b1c2 - c1b2) k] · (a1i + a2j + a3k)
a1(b2c3 - b3c2 )- a2(b3c1 - b1c3) + a3(b1c2 - b2c1)
Geometrical Interpretation of Scalar Triple Product
The scalar triple product (a b × c)represents the signed volume of the parallelepiped formed by three vectors (a, b, c) as adjacent edges. The base of the parallelepiped is defined by the vectors b and c, their cross product b × c, gives a vector perpendicular to this base, and its magnitude equals the area of the base. The vector a is the height, (b × c) · a is the volume of the parallelepiped:
|a · (b × c)| = |b × c| · |a| · cosθ, where θ is the angle between the vector a and b × c.
Real-world Applications of Scalar Triple Product
The scalar triple product is used in various fields such as mathematics, physics, geometry, and computer graphics. In this section, we will explore some real-life applications of scalar triple products.
- In engineering and CAD, the scalar triple product calculates the volume of parallelepipeds, such as slanted roofs in architectural designs. For example, the architecture students use the scalar triple product to find the volume of irregular rooms or slanted structures.
- In GPS and aviation, the scalar triple product is used to navigate and to determine the orientation of planes or satellites. It also helps in checking the co-planarity or computing the spatial volume formed by vectors.
- In physics, the scalar triple product is used to compute the volume of a region, magnetic flux, and used in fluid dynamics.
Common Mistakes and How to Avoid Them in Scalar Triple Product
We use the scalar triple product for vector algebra calculations. However, students often make mistakes due to confusion in operations, misinterpreting properties, or calculation errors. Here, we will discuss the common mistakes and how to avoid them.
Mistake 1
Confusing scalar and vector triple product
Using the scalar triple product for a vector triple product leads to errors, as the results of the two operations are different. Remember that the scalar triple product is calculated by taking the dot product of a vector with the cross product of the other two vectors.
Mistake 2
Error while performing the cross product
Errors are common among students when performing cross products, such as incorrect signs and misapplying the cross product formula. To avoid these errors, use the 2 × 2 determinant method for each component systematically, and also double-check the sign.
Mistake 3
Miscalculating the dot product
When performing the dot product, students make errors by using incorrect components or misaligning terms. So, when performing the scalar triple product, first find the cross product result before calculating the dot product and match each component carefully. Then calculate each multiplication separately and then add the results step by step.
Mistake 4
Using the scalar triple product for a two-dimensional vector
Students sometimes use the scalar triple product with 2D vectors such as b = (b1, b2), but this is wrong as the scalar triple product is applicable only for 3D, not for 2D. So always make sure that the vector is in 3D, not in 2D, and extend it by adding 0. For example, b = (b1, b2, 0).
Mistake 5
Thinking that the result is always positive
Students assume that the result is always positive, but it is not true in all cases. So remember that the sign depends on the vectors’ orientation, the negative results indicate the left-handed orientation, and zero indicates co-planarity.
Solved Examples of Scalar Triple Product
Problem 1
Find the scalar triple product of the vectors: i, j, k
i (j × k) = 1
Explanation
Here, we first find the cross product of j and k and then their dot product with i.
Cross product: j × k = i
Dot product: i i = 1
Problem 2
Determine the scalar triple product of the vectors i + j, 2i + 2j, i - j
The scalar triple product is 0
Explanation
Given,
A =i + j
B =2i + 2j
C = i - j
Writing in component form:
A = (1, 1, 0)
B = (2, 2, 0)
C = (1, -1, 0)
Finding B × C
= i (2 0 - 0 (-1)) - j (2 0 - 0 1) + k (2 (-1) - 2 1)
= 0i - 0j + (-4)k
= -4k
Finding the dot product with A
A (B × C) = (1, 1, 0) · (0, 0, -4) = 1 · 0 + 1 · 0 + 0 · -4
= 0
Problem 3
Calculate the volume of the parallelepiped formed by the vectors A = (2, -1, 3), B =4, 0, -2 and C = -1, 2, 1
The volume of the parallelepiped is 34 cubic units.
Explanation
Finding the scalar triple product using the formula:
A (B × C) =
= 2(0 · 1 - (-2) · 2) - (-1)(4 · 1 - (-2) (-1)) + 3(4 · 2 - 0 · (-1))
= 8 + 2 + 24 = 34.
Problem 4
Find the scalar triple product of the vectors, A = (3, 0, -1), B = (-2, 5, 1), and C = (4, 1, -3)
The scalar triple product is -26
Explanation
The formula to find the scalar triple product is:
A (B × C) =
= 3 (5 · (-3) - 1 · 1) - 0(-2 · (-3) - 1 · 4) + (-1)(-2 · 1 - 5 · 4)
= 3(-16) + 0 + (-1)(-2 - 20)
= -48 + 22
= -26
Problem 5
Determine the volume of a parallelepiped whose adjacent edges are A = i + j + k C = i + j - k
The volume of a parallelepiped is 4 cubic units.
Explanation
To find the volume of a parallelepiped, we use the formula:
A = i + j + k = (1, 1, 1)
B = i - j + k = (1, -1, 1)
C = i + j - k = (1, 1, -1)
A (B × C) =
= 1((-1)(-1) - 1 · 1) - 1(1 · (-1) - 1 · 1) + 1(1 · 1 - (-1) · 1)
= 1(0) - 1(-2) + 1(2)
= 0 - (-2) + 2
= 4
FAQs on Scalar Triple Product
1.What is the scalar triple product?
2.What is the formula for the scalar triple product?
3.Is the scalar product always zero?
4.Can a scalar triple product be negative?
5.What if the scalar triple product is zero?
6.How does learning Algebra help students in Canada make better decisions in daily life?
7.How can cultural or local activities in Canada support learning Algebra topics such as Scalar Triple Product?
8.How do technology and digital tools in Canada support learning Algebra and Scalar Triple Product?
9.Does learning Algebra support future career opportunities for students in Canada?
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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