Vector Triple Product

In vector algebra, the vector triple product is a mathematical operation that combines three vectors to produce a new vector. It is an extension of the cross product of two vectors. The vector triple product is used in various areas of physics and engineering, including mechanics, electromagnetism, and quantum mechanics.

Definition

The vector triple product is defined as the cross product of two vectors, followed by another cross product with a third vector. Mathematically, it can be expressed as:

$$\mathbf{A} \times (\mathbf{B} \times \mathbf{C})$$

where $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ are vectors.

Calculation

To calculate the vector triple product, we can use the following formula:

$$\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B})$$

where $\mathbf{A} \cdot \mathbf{B}$ represents the dot product of vectors $\mathbf{A}$ and $\mathbf{B}$.

Properties

The vector triple product has several important properties that are useful in vector algebra and physics. Some of these properties include:

1. Associativity: The vector triple product is associative, which means that the order of the vectors does not matter. In other words, $(\mathbf{A} \times \mathbf{B}) \times \mathbf{C} = \mathbf{A} \times (\mathbf{B} \times \mathbf{C})$.

2. Distributivity: The vector triple product is distributive over addition, which means that $(\mathbf{A} + \mathbf{B}) \times \mathbf{C} = \mathbf{A} \times \mathbf{C} + \mathbf{B} \times \mathbf{C}$.

3. Scalar multiplication: The vector triple product can be multiplied by a scalar, which means that $k(\mathbf{A} \times \mathbf{B} \times \mathbf{C}) = (k\mathbf{A}) \times (\mathbf{B} \times \mathbf{C}) = \mathbf{A} \times (k\mathbf{B} \times \mathbf{C}) = \mathbf{A} \times (\mathbf{B} \times k\mathbf{C})$.

4. Orthogonality: The vector triple product is orthogonal to all three vectors involved. This means that $(\mathbf{A} \times \mathbf{B} \times \mathbf{C}) \cdot \mathbf{A} = (\mathbf{A} \times \mathbf{B} \times \mathbf{C}) \cdot \mathbf{B} = (\mathbf{A} \times \mathbf{B} \times \mathbf{C}) \cdot \mathbf{C} = 0$.

Examples

Let's consider some examples to understand the vector triple product better.

Example 1:

Given vectors $\mathbf{A} = 2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}$, $\mathbf{B} = \mathbf{i} + 2\mathbf{j} - \mathbf{k}$, and $\mathbf{C} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}$, calculate $\mathbf{A} \times (\mathbf{B} \times \mathbf{C})$.

Using the formula, we have:

$$\mathbf{B} \times \mathbf{C} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 2 & -1 \ 3 & -1 & 2 \end{vmatrix} = 5\mathbf{i} + 7\mathbf{j} + 5\mathbf{k}$$

$$\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B})$$

Calculating the dot products:

$$\mathbf{A} \cdot \mathbf{C} = (2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}) \cdot (3\mathbf{i} - \mathbf{j} + 2\mathbf{k}) = 6 + 3 - 8 = 1$$

$$\mathbf{A} \cdot \mathbf{B} = (2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}) \cdot (\mathbf{i} + 2\mathbf{j} - \mathbf{k}) = 2 + (-6) + (-4) = -8$$

Substituting the values:

$$\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = (5\mathbf{i} + 7\mathbf{j} + 5\mathbf{k})(1) - (3\mathbf{i} - \mathbf{j} + 2\mathbf{k})(-8)$$ $$= 5\mathbf{i} + 7\mathbf{j} + 5\mathbf{k} + 24\mathbf{i} - 8\mathbf{j} - 16\mathbf{k}$$ $$= 29\mathbf{i} - \mathbf{j} - 11\mathbf{k}$$

Therefore, $\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = 29\mathbf{i} - \mathbf{j} - 11\mathbf{k}$.

Example 2:

Given vectors $\mathbf{A} = \mathbf{i} + 2\mathbf{j} - \mathbf{k}$, $\mathbf{B} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}$, and $\mathbf{C} = 2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}$, calculate $(\mathbf{A} \times \mathbf{B}) \times \mathbf{C}$.

Using the formula, we have:

$$\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 2 & -1 \ 3 & -1 & 2 \end{vmatrix} = 5\mathbf{i} + 7\mathbf{j} + 5\mathbf{k}$$

$$(\mathbf{A} \times \mathbf{B}) \times \mathbf{C} = \mathbf{A}(\mathbf{B} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B})$$

Calculating the dot products:

$$\mathbf{B} \cdot \mathbf{C} = (3\mathbf{i} - \mathbf{j} + 2\mathbf{k}) \cdot (2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}) = 6 - 3 - 8 = -5$$

$$\mathbf{A} \cdot \mathbf{B} = (\mathbf{i} + 2\mathbf{j} - \mathbf{k}) \cdot (3\mathbf{i} - \mathbf{j} + 2\mathbf{k}) = 3 - 2 - 2 = -1$$

Substituting the values:

$$(\mathbf{A} \times \mathbf{B}) \times \mathbf{C} = (5\mathbf{i} + 7\mathbf{j} + 5\mathbf{k})(-5) - (2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k})(-1)$$ $$= -25\mathbf{i} - 35\mathbf{j} - 25\mathbf{k} + 2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}$$ $$= -23\mathbf{i} - 32\mathbf{j} - 29\mathbf{k}$$

Therefore, $(\mathbf{A} \times \mathbf{B}) \times \mathbf{C} = -23\mathbf{i} - 32\mathbf{j} - 29\mathbf{k}$.

Summary

The vector triple product is a mathematical operation that combines three vectors to produce a new vector. It is calculated by taking the cross product of two vectors, followed by another cross product with a third vector. The vector triple product has several properties, including associativity, distributivity, scalar multiplication, and orthogonality. It is used in various areas of physics and engineering to solve problems involving vector quantities.