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Scalar Triple Product: Definition, Formula, and Examples

Scalar triple product is an important concept in vector algebra, this mathematical operation involve three vectors in three-dimensional space, denoted as [Tex]\mathbf{a}[/Tex], [Tex]\mathbf{b}[/Tex], and [Tex]\mathbf{c}[/Tex]. It is defined as the dot product of one of the vectors with the cross product of the other two vectors.

What is Scalar Triple Product?

Scalar triple product is a mathematical operation involving three vectors in three-dimensional space. It is defined as the dot product of one vector with the cross product of the other two vectors. Given vectors A, B, and C, the scalar triple product is expressed as A · (B × C). The result is a scalar value, not a vector. This value represents the volume of the parallelepiped formed by the three vectors. The scalar triple product is significant in various applications, including physics and engineering, where it helps determine volumes and solve problems related to vector geometry. Mathematically, it is expressed as:

a . (b x c)

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Geometrical Interpretation of Scalar Triple Product

The scalar triple product vectors A · (B × C) yields a scalar value that represents the volume of the parallelepiped formed by these vectors.

A parallelepiped is a three-dimensional figure with six parallelogram faces. The vectors B and C form a parallelogram base, and the vector A extends perpendicular to this base. The magnitude of the cross product B × C gives the area of the parallelogram formed by B and C. When this area is dotted with vector A, the resulting scalar represents the height of the parallelepiped perpendicular to the base. Thus, the scalar triple product measures the volume of the parallelepiped, with its sign indicating the orientation (right-handed or left-handed) of the vector triplet.

Scalar Triple Product Formula

The formula for the scalar triple product involves three vectors A, B, and C in three-dimensional space. The scalar triple product is given by:

A · (B × C) = [Tex]\begin{vmatrix} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \\ \end{vmatrix}[/Tex] = Ax(ByCz) + Ay(BzCx) + Az(BxCy) – Ax(BzCy) – Ay(BxCz) – Az(ByCx)

This determinant provides a concise and compact way to calculate the scalar triple product.

Proof of the Formula of Scalar Triple Product

The proof of the scalar triple product formula involves demonstrating that the scalar triple product can be expressed as the determinant of a 3×3 matrix formed by the components of the vectors. Here’s a step-by-step proof:

Given vectors A, B, and C with components:

  • A = (Ax, Ay, Az)
  • B = (Bx, By, Bz)
  • C = (Cx, Cy, Cz)

First, calculate the cross product B × C:

B × C = [Tex]\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ B_x & B_y & B_z \\ C_x & C_y & C_z \\ \end{vmatrix}[/Tex]

⇒ B × C = (ByCz – BzCy, BzCx – BxCz, BxCy – ByCx)

Next, take the dot product of A with the resulting vector:

A · (B × C) = (Ax, Ay, Az) · (ByCz – BzCy, BzCx – BxCz, BxCy – ByCx)

= Ax(ByCz – BzCy) + Ay(BzCx – BxCz) + Az(BxCy – ByCx)

Now, expand the dot product:

A · (B × C) = Ax(ByCz) – Ax(BzCy) + Ay(BzCx) – Ay(BxCz) + Az(BxCy) – Az(ByCx)

Group the terms to form a determinant:

A · (B × C) = Ax(ByCz) + Ay(BzCx) + Az(BxCy) – Ax(BzCy) – Ay(BxCz) – Az(ByCx)

This can be written as the determinant of a 3×3 matrix:

A · (B × C) = [Tex]\begin{vmatrix} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \\ \end{vmatrix}[/Tex]

To verify, let’s evaluate the determinant explicitly:

[Tex]\begin{vmatrix} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \\ \end{vmatrix}[/Tex] = Ax(ByCz – BzCy) – Ay(BxCz – BzCx) + Az(BxCy – ByCx)

Thus, the scalar triple product A · (B × C) is indeed equal to the determinant of the 3×3 matrix formed by the components of vectors A, B, and C. This proves the formula.

Properties of Scalar Triple Product

  • Anti-symmetry: The scalar triple product changes sign if any two vectors are swapped:

A · (B × C) = -A · (C × B) = B · (C × A) = -B · (A × C) = C · (A × B) = -C · (B × A)

  • Linearity: The scalar triple product is linear with respect to each vector:

A · (B × C) = (A · B) · C – (A · C) · B

  • Cyclic Permutation: The scalar triple product remains unchanged under cyclic permutations of the vectors:

A · (B × C) = B · (C × A) = C · (A × B)

  • Volume Interpretation: The scalar triple product represents the signed volume of the parallelepiped formed by vectors A, B, and C. The sign indicates the orientation (right-handed or left-handed) of the vector triplet.
  • Orthogonality: If the scalar triple product is zero (A · (B × C) = 0), then vectors A, B, and C are coplanar.

Scalar Triple Product vs Vector Triple Product

The key difference between scaler and vector triple product are listed in the following table:

Feature

Scalar Triple Product

Vector Triple Product

Representation

A · (B × C), where A, B, C are vectors

A × (B × C), where A, B, C are vectors

Result

Scalar (a number)

Vector (directional result)

Geometric Interpretation

Volume of parallelepiped spanned by A, B, and C vectors

Vector representing the directed volume of parallelepiped

Anti-symmetry

Yes, changes sign with permutation of vectors

No, does not change sign with permutation of vectors

Applications

Volume calculations, orientation tests in 3D

Rotation and moment calculations, advanced vector transformations

Conclusion

The scalar triple product is a fundamental concept in vector algebra with versatile applications in geometry, physics, and engineering. It provides insights into spatial volumes, orientations, and coplanarity of vectors in three-dimensional space, essential for solving various theoretical and practical problems in mathematical analysis and applied sciences.

Sample Problems on Scalar Triple Product: Solved

Problem 1: Calculating Volume Given vectors A = (2, -1, 3), B = (4, 0, -2), and C = (-1, 2, 1), calculate the volume of the parallelepiped formed by these vectors.

Solution:

To find the volume V of the parallelepiped, use the formula |A · (B × C)|.

[Tex]B \times C = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 4 & 0 & -2 \\ -1 & 2 & 1 \end{vmatrix}[/Tex]

Expanding this determinant, we get:

[Tex]B \times C = \mathbf{i}(0 \cdot 1 – (-2) \cdot 2) – \mathbf{j}(4 \cdot 1 – (-2) \cdot (-1)) + \mathbf{k}(4 \cdot 2 – 0 \cdot (-1))[/Tex]

[Tex]\Rightarrow B \times C = \mathbf{i}(0 + 4) – \mathbf{j}(4 – 2) + \mathbf{k}(8 – 0)[/Tex]

[Tex]\Rightarrow B \times C = 4\mathbf{i} – 2\mathbf{j} + 8\mathbf{k}[/Tex] = (4, -2, 8)

Next, we calculate the dot product of A with the result of B \times C:

[Tex]A \cdot (B \times C) = (2, -1, 3) \cdot (4, -2, 8)[/Tex]

[Tex]\Rightarrow A \cdot (B \times C) = 2 \cdot 4 + (-1) \cdot (-2) + 3 \cdot 8[/Tex]

[Tex]\Rightarrow A \cdot (B \times C) = 8 + 2 + 24[/Tex] = 34

Therefore, the volume of the parallelepiped formed by vectors A, B, and C is 38 cubic units.

Problem 2: Testing Coplanarity Determine if vectors A = (1, 2, 3), B = (4, 5, 6), and C = (7, 8, 9) are coplanar.

Solution:

Vectors A, B, and C are coplanar if A · (B × C) = 0.

[Tex]B \times C = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix}[/Tex]

This can be expanded as:

[Tex]B \times C = \mathbf{i}(5 \cdot 9 – 6 \cdot 8) – \mathbf{j}(4 \cdot 9 – 6 \cdot 7) + \mathbf{k}(4 \cdot 8 – 5 \cdot 7)[/Tex]

⇒ [Tex]B \times C = \mathbf{i}(45 – 48) – \mathbf{j}(36 – 42) + \mathbf{k}(32 – 35)[/Tex]

⇒ [Tex]B \times C = \mathbf{i}(-3) – \mathbf{j}(-6) + \mathbf{k}(-3)[/Tex] = (-3, 6, -3)

Next, we calculate the dot product of A with the result of B \times C:

[Tex]A \cdot (B \times C) = (1, 2, 3) \cdot (-3, 6, -3)[/Tex]

⇒ [Tex]A \cdot (B \times C) = 1 \cdot (-3) + 2 \cdot 6 + 3 \cdot (-3)[/Tex] = -3 + 12 – 9 = 0

Since A · (B × C) = 0, vectors A, B, and C are coplanar.

Practice Problems on Scalar Triple Product: Unsolved

Problem 1: Given vectors A, B, and C in three-dimensional space, calculate the volume of the parallelepiped formed by these vectors using the scalar triple product.

Problem 2: Determine if the vectors A, B, and C are coplanar by evaluating the scalar triple product. Vectors are coplanar if the scalar triple product equals zero.

Scalar Triple Product – FAQs

What is the scalar triple product?

The scalar triple product is a mathematical operation involving three vectors in three-dimensional space. It is defined as the dot product of one vector with the cross product of the other two vectors.

How is the scalar triple product used?

It is used to calculate volumes of parallelepipeds and determine the orientation of vectors in three-dimensional space. It also helps in solving problems related to geometry, physics, and engineering.

What does a scalar triple product of zero indicate?

A scalar triple product of zero indicates that the three vectors involved are coplanar, meaning they lie on the same plane in space.

What is the geometrical interpretation of the scalar triple product?

Geometrically, the scalar triple product represents the signed volume of the parallelepiped formed by the three vectors. The sign indicates the orientation (right-handed or left-handed) of the vector triplet.


 


Reffered: https://www.geeksforgeeks.org

Mathematics

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