# Triple Product Calculator

## Calculate triple products step by step

The calculator will calculate the triple product (both scalar and vector) of the three vectors, with steps shown.

Welcome to our Triple Product Calculator, an invaluable online tool that simplifies calculating the triple product (both scalar and vector) of vectors.

In the field of linear algebra and vector analysis, the triple product refers to the scalar result from the operation of two cross products and a dot product of three vectors, generally represented as $$$\mathbf{\vec{u}}\cdot\left(\mathbf{\vec{v}}\times\mathbf{\vec{w}}\right)$$$ or $$$\left(\mathbf{\vec{u}}\times\mathbf{\vec{v}}\right)\cdot\mathbf{\vec{w}}$$$. This concept is pivotal in various applications such as calculating the volume of a parallelepiped.

## How to Use the Triple Product Calculator?

### Input

Firstly, you need to input the three vectors for which you want to calculate the triple product.

### Calculation

Once you've entered the vectors, click on the "Calculate" button. The calculator will automatically compute the triple products (both scalar and vector).

### Result

After clicking on the button, the result will be displayed on the screen. This result is a scalar value that represents the scalar triple product of your three vectors, as well as vectors that represent the vector triple products.

## Understanding the Triple Product of Vectors

In vector analysis, the triple product of vectors is an operation that combines the cross product and the dot product.

Given three vectors $$$\mathbf{\vec{u}}$$$, $$$\mathbf{\vec{v}}$$$, and $$$\mathbf{\vec{w}}$$$ in three-dimensional space, the triple product is defined by one of the following equivalent formulas: $$$\mathbf{\vec{u}}\cdot\left(\mathbf{\vec{v}}\times\mathbf{\vec{w}}\right)$$$ or $$$\left(\mathbf{\vec{u}}\times\mathbf{\vec{v}}\right)\cdot\mathbf{\vec{w}}$$$.

Here, $$$\cdot$$$ denotes a dot product and $$$\times$$$ denotes a cross product.

The value of the triple product $$$\mathbf{\vec{u}}\cdot\left(\mathbf{\vec{v}}\times\mathbf{\vec{w}}\right)$$$ gives the volume of the parallelepiped formed by the three vectors $$$\mathbf{\vec{u}}$$$, $$$\mathbf{\vec{v}}$$$, and $$$\mathbf{\vec{w}}$$$. If the vectors $$$\mathbf{\vec{u}}$$$, $$$\mathbf{\vec{v}}$$$, and $$$\mathbf{\vec{w}}$$$ are coplanar, then $$$\mathbf{\vec{u}}\cdot\left(\mathbf{\vec{v}}\times\mathbf{\vec{w}}\right)=0$$$, as the volume of the parallelepiped is zero.

Let's illustrate this with an example: consider three vectors $$$\mathbf{\vec{u}}=\langle1,2,3\rangle$$$, $$$\mathbf{\vec{v}}=\langle4,5,6\rangle$$$, and $$$\mathbf{\vec{w}}=\langle7,8,9\rangle$$$.

The cross product of the vectors $$$\mathbf{\vec{v}}$$$ and $$$\mathbf{\vec{w}}$$$ (i.e., $$$\mathbf{\vec{v}}\times\mathbf{\vec{w}}$$$) can be computed using the determinant of a 3x3 matrix:

$$\mathbf{\vec{v}}\times\mathbf{\vec{w}}=\langle5\cdot9-6\cdot8,-(4\cdot9-6\cdot7),4\cdot8-5\cdot7\rangle=\langle-3,6,-3\rangle$$

Now, you compute the dot product of $$$\mathbf{\vec{u}}$$$ and the cross product $$$\mathbf{\vec{v}}\times\mathbf{\vec{w}}$$$ (i.e., $$$\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}\times\mathbf{\vec{w}}$$$):

$$\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}\times\mathbf{\vec{w}}=1\cdot(-3)+2\cdot6+3\cdot(-3)=0$$Hence, the triple product of the vectors $$$\mathbf{\vec{u}}$$$, $$$\mathbf{\vec{v}}$$$, and $$$\mathbf{\vec{w}}$$$ is $$$0$$$. This can be calculated easily using our triple product calculator.

Remember, the order of vectors matters in the triple product calculation. Changing the order can result in a different sign of the output.

## Why Choose Our Triple Product Calculator?

### User-Friendly

Our calculator is designed to be simple and easy to use, making it accessible even for those who are new to vector calculations.

### Time-Efficient

Manual calculations, especially for complex vectors, can be time-consuming. Our calculator instantly computes the triple product, saving you valuable time.

### Accurate

Our Triple Product Calculator ensures highly accurate results, reducing the risk of errors that can occur with manual calculations.

### Available

As an online tool, our calculator is available 24/7 and can be accessed from anywhere with an internet connection.

### Educational

It's not just a tool for getting quick answers. It can also be a helpful resource for students to understand the process of calculating the triple product of vectors.

### FAQ

#### What is a triple product of vectors?

A triple product of vectors is a scalar result from the operation of two cross products and a dot product of three vectors. It is represented as $$$\mathbf{\vec{u}}\cdot\left(\mathbf{\vec{v}}\times\mathbf{\vec{w}}\right)$$$ or $$$\left(\mathbf{\vec{u}}\times\mathbf{\vec{v}}\right)\cdot\mathbf{\vec{w}}$$$.

#### What is the significance of the triple product in real-world applications?

The triple product of vectors has several real-world applications. It is used in physics and engineering to calculate the volume of a parallelepiped, for instance.

#### What does the Triple Product Calculator do?

The Triple Product Calculator is an online tool that helps you calculate the triple product of three vectors. It simplifies the complex calculation process and gives you accurate results quickly.