Vector Triple Product

The vector triple product refers to the cross product of one vector with the cross product of two other vectors. In mathematical terms, if we have three vectors A, B, and C, the vector triple product is given by A × (B × C).

Formula

The vector triple product A × (B × C) can be expanded using the following formula:

A × (B × C) = (A · C)B - (A · B)C

Here, "×" denotes the cross product and "·" denotes the dot product.

Properties

The vector triple product has several important properties:

1. Non-commutativity: The cross product is not commutative, which means that A × (B × C) is not the same as B × (A × C).
2. Distributivity over addition: The cross product is distributive over vector addition, i.e., A × (B + C) = A × B + A × C.
3. Scalar multiplication: Scalar multiplication is distributive over the cross product, i.e., k(A × B) = (k*A) × **B* = A × (k*B*) for any scalar k.

Differences and Important Points

Property	Description
Direction	The result of a vector triple product is a vector that is perpendicular to the original vector A.
Magnitude	The magnitude of the vector triple product is equal to the volume of the parallelepiped formed by A, B, and C.
Dependency	The result of the vector triple product depends on the order of the vectors due to the non-commutativity of the cross product.

Examples

Example 1: Calculation of Vector Triple Product

Let's calculate the vector triple product of vectors A = (1, 2, 3), B = (4, 5, 6), and C = (7, 8, 9).

First, we find B × C:

B × C = |i j k| |4 5 6| |7 8 9|

= i(5×9 - 6×8) - j(4×9 - 6×7) + k(4×8 - 5×7) = i(45 - 48) - j(36 - 42) + k(32 - 35) = -3i + 6j - 3k

Now, we find A × (B × C):

A × (B × C) = |i j k| |1 2 3| |-3 6 -3|

= i(2×(-3) - 3×6) - j(1×(-3) - 3×(-3)) + k(1×6 - 2×(-3)) = -24i + 6j + 12k

Therefore, the vector triple product A × (B × C) is (-24, 6, 12).

Example 2: Using the Vector Triple Product Formula

Given vectors A = (1, 0, 0), B = (0, 1, 0), and C = (0, 0, 1), let's use the formula to find A × (B × C).

Using the formula:

A × (B × C) = (A · C)B - (A · B)C

Since A · C = 0 and A · B = 0 (because A, B, and C are orthogonal), we get:

A × (B × C) = (0)B - (0)C = (0, 0, 0)

So, the vector triple product in this case is the zero vector.

Example 3: Geometric Interpretation

Consider vectors A, B, and C representing the edges of a parallelepiped. The volume of the parallelepiped can be found using the scalar triple product |A · (B × C)|. The vector triple product A × (B × C) gives a vector that is perpendicular to A and has a magnitude equal to the volume of the parallelepiped.

Conclusion

The vector triple product is a fundamental concept in vector calculus and physics, especially in areas involving torque, angular momentum, and volumes in three-dimensional space. Understanding its properties and being able to compute it are essential skills for students and professionals in mathematics, physics, and engineering.