Vector Triple Product

The vector triple product involves the multiplication of three vectors so that the output is also a vector. The vector triple product involves three vectors— \vec{a}, \vec{b}, and \vec{c}. By taking the cross product of \vec{a} with the cross product of \vec{b} and \vec{c} the result, denoted as \vec{a} \times (\vec{b} \times \vec{c}), emerges as a new vector.

The vector triple product involves three vectors: \vec{a}, \vec{b}, and \vec{c}. It is the result of taking the cross product of \vec{a} with the cross product of \vec{b} and \vec{c}. Mathematically, it's expressed as \vec{a} \times (\vec{b} \times \vec{c} )

The resulting vector lies in the plane of \vec{b} and \vec{c}. Another way to represent the vector triple product is by expressing it as a combination of \vec{b} and \vec{c}, written as \vec{a} × (\vec{b} × \vec{c}) = x \vec{b} + y \vec{c}).

Vector Triple Product Formula

The vector triple product involves three vectors: \vec{a}, \vec{b}, and \vec{c}. The formula is \vec{a} \times (\vec{b} \times \vec{c}).

Cross Product of \vec{b} and \vec{c}: First, find the cross product of \vec{b} and \vec{c}.

\vec{b} \times \vec{c}

Multiply by \vec{a}: Take this result and perform a cross product with \vec{a}.

\vec{a} \times (\vec{b} \times \vec{c})

Resulting Vector: The final vector obtained is coplanar with \vec{b} and \vec{c} and perpendicular to \vec{a}.

This formula can also be expressed as a linear combination of \vec{b} and \vec{c}:

\vec{a} \times (\vec{b} \times \vec{c}) = x \vec{b} + y \vec{c}

This means the triple product result can be written as a combination of \vec{b} and \vec{c}, where (x) and (y) are coefficients.

Now, the vector triple product formula is,

\vec{a}\times(\vec{b}\times\vec{c})~=~(\vec{a}.\vec{c})\vec{b}~-~(\vec{a}.\vec{b})\vec{c}

Vector Triple Product Proof

Proving the vector triple product formula involves some vector algebra. Let's break it down step by step

The vector triple product formula is \vec{a} \times (\vec{b} \times \vec{c})

Start with Cross Product

\vec{b} \times \vec{c}

Use Scalar Triple Product Identity

The scalar triple product identity is \vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot (\vec{c} \times \vec{a}). Applying this identity, we can rewrite the expression:

\vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot (\vec{c} \times \vec{a})

Expand both sides of the equation using the vector triple product definition:

\vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot (\vec{c} \times \vec{a})

\vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot (\vec{a} \times \vec{c})

Apply Vector Triple Product Identity

The vector triple product identity is \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}

Substitute this into the equation:

(\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c} = \vec{b} \cdot (\vec{a} \times \vec{c})

Rearrange the terms to isolate (\vec{a} \times (\vec{b} \times \vec{c}):

\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}

Properties of Vector Triple Product

Properties of the Vector Triple Product are,

• Vector Nature: The vector triple product is itself a vector quantity.
• Unit Vector: There exists a unit vector that lies in the same plane as \vec{a} \vec{a} and \vec{b}, and is perpendicular to \vec{c}. This unit vector is given by \pm \frac{(\vec{a} \times \vec{b}) \times \vec{c}}{\lvert (\vec{a} \times \vec{b}) \times \vec{c} \rvert}
• Distinct from Cross Products: It's crucial to understand that \vec{a} \times (\vec{b} \times \vec{c}) is not equal to (\vec{a} \times \vec{b}) \times \vec{c}.
• Non-Coplanar Vectors: If \vec{a}, \vec{b}, and \vec{c} are non-coplanar vectors (they do not lie in the same plane), then \vec{a} \times \vec{b} , \vec{b} \times \vec{c}, and \vec{c} \times \vec{a} are also non-coplanar.

The cross product is not associative

• For vectors a, b, and c, a×(b×c) ≠ (a×b)×c.

Article Related to Vector Triple Product:

• Vector Operations
• Vector Addition
• Scalar and Vector
• Dot and Cross Product of Vectors

Examples on Vector Triple Product

Example 1: Given three vectors \vec{a} = \hat{i} + 2\hat{j} - \hat{k}, \vec{b} = \hat{i} - \hat{j} + \hat{k}, and \vec{c} = \hat{i} + \hat{j} + \hat{k}, calculate the vector triple product \vec{a} \times (\vec{b} \times \vec{c})

Solution: Find \vec{b} \times \vec{c}

\vec{b} \times \vec{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -1 & 1 \\ 1 & 1 & 1 \end{vmatrix} = 2\hat{i} - 2\hat{j}

Multiply by \vec{a}

\vec{a} \times (\vec{b} \times \vec{c}) = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & -1 \\ 2 & -2 & 0 \end{vmatrix} = -2\hat{i} + 4\hat{j} + 6\hat{k}

Therefore, \vec{a} \times (\vec{b} \times \vec{c}) = -2\hat{i} + 4\hat{j} + 6\hat{k}

Example 2: Verify whether the equation \vec{p} = \vec{q} \times (\vec{r} \times \vec{s}) holds true, where \vec{p} = \hat{i} + \hat{j}, \vec{q} = 2\hat{i} - \hat{j} , \vec{r} = \hat{i} + 3\hat{j} + 2\hat{k} , and \vec{s} = \hat{k}.

Solution: Calculate \vec{r} \times \vec{s}:

\vec{r} \times \vec{s} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 3 & 2 \\ 0 & 0 & 1 \end{vmatrix} = -\hat{j} + \hat{k}

Multiply by \vec{q}

\vec{q} \times (\vec{r} \times \vec{s}) = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -1 & 0 \\ -1 & 1 & 0 \end{vmatrix} = \hat{k}

Compare with \vec{p}

\vec{p} = \hat{i} + \hat{j} and \vec{q} \times (\vec{r} \times \vec{s}) = \hat{k}

Since \vec{p} and \vec{q} \times (\vec{r} \times \vec{s}) are not equal, the equation \vec{p} = \vec{q} \times (\vec{r} \times \vec{s}) does not hold true.

Practice Problems

Problem 1. Given three vectors \vec{a}, \vec{b}, \vec{c}, calculate the vector triple product \vec{a} \times (\vec{b} \times \vec{c}).

Problem 2. Determine the unit vector that is coplanar with \vec{u} = 3\hat{i} + 2\hat{j} - \hat{k} and \vec{v} = \hat{i} - \hat{j} + 2\hat{k}, and perpendicular to \vec{w} = 2\hat{i} + \hat{j} + 3\hat{k} using the vector triple product.

Problem 3. Verify whether the equation \vec{p} = \vec{q} \times (\vec{r} \times \vec{s}) holds true, where \vec{p} = \hat{i} + \hat{j}, \vec{q} = 2\hat{i} - \hat{j}, \vec{r} = \hat{i} + 3\hat{j} + 2\hat{k}, and \vec{s} = \hat{k}.

Problem 4. If \vec{a} = 2\hat{i} - \hat{j}, \vec{b} = \hat{i} + \hat{j} + \hat{k}, and \vec{c} = -\hat{j} + 3\hat{k}, find the angle between \vec{a} \times (\vec{b} \times \vec{c}) and \vec{c}.

Problem 5. Given non-coplanar vectors \vec{u} = \hat{i} - 2\hat{j} + 3\hat{k}, \vec{v} = 2\hat{i} + \hat{j} - \hat{k}, and \vec{w} = -\hat{i} + \hat{j} + 2\hat{k}, prove that \vec{u} \times \vec{v}, \vec{v} \times \vec{w}, and \vec{w} \times \vec{u} are also non-coplanar.