Vector Triple Product

Last Updated : 14 Dec, 2023

Vector Triple Product involves the multiplication of three vectors so that the output is also a vector. 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.

This article covers the definition, formula, proof, and properties of the Vector Triple Product, offering a comprehensive exploration of its fundamental aspects. Additionally, we will address common questions and provide solved examples to enhance your understanding of this mathematical concept.

Table of Content

Vector Triple Product Definition
Vector Triple Product Formula
Vector Triple Product Proof
Properties of Vector Triple Product
Examples on Vector Triple Product

Vector Triple Product Definition

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 $\vec{a}$ × $\vec{b}$ × $\vec{c}$ . The resulting vector lies in the same plane as $\vec{b}$ and $\vec{c}$ and is perpendicular to $\vec{a}$ . 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 the Cross Product

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

Use the 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 ( $\vec{a} \times (\vec{b} \times \vec{c}$ ):

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

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 $\vec{a}$ 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.

Associative Property of Vector Triple Product

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

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 on Vector Triple Product

P1. Given three vectors $\vec{a}$ , $\vec{b}$ , $\vec{c}$ , calculate the vector triple product $\vec{a} \times (\vec{b} \times \vec{c})$ .

P2. 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.

P3. 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 $\vec{s} = \hat{k}$ .

P4. 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 $\vec{c}$ .

P5. 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.

Vector Triple Product-FAQs

1. What is Vector Triple Product?

Vector Triple Product is a mathematical operation involving three vectors. Specifically, it refers to the cross product of the cross product of two vectors, providing a new vector as the result.

2. How is the Vector Triple Product Expressed Mathematically?

Mathematically, the Vector Triple Product involving vectors A, B, and C is denoted as A×(B×C).

3. What is the Geometric Interpretation of Vector Triple Product?

Geometrically, the Vector Triple Product is associated with a parallelepiped, a six-faced figure formed by three vectors. The resulting vector represents the normal to the parallelepiped’s face.

4. What is the Formula for Triple Dot Product?

For three vectors a, b, and c the vector triple product is written [a, b, c] and is calculated as,

[a, b, c] = a×(b×c)

5. What is the Vector Triple Product Identity?

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

Suggest improvement

Product of Vectors

Share your thoughts in the comments