# Dot Product Calculator

## Find the dot product of two vectors step by step

An online calculator for finding the dot (inner) product of two vectors, with steps shown.

The Dot Product Calculator is an accessible online resource that performs dot product calculations. With its user-friendly interface, it makes complex computations effortless. Ideal for students, educators, and professionals involved in vector mathematics, this tool is designed to make your calculations straightforward and accurate.

## How to Use the Dot Product Calculator?

### Input

In the provided fields, input the coordinates for the two vectors for which you wish to calculate the dot product. If your vectors are in the $$$\mathbf{\vec{i}}$$$, $$$\mathbf{\vec{j}}$$$, $$$\mathbf{\vec{k}}$$$ format, enter them in that format or convert them into the coordinate form $$$\langle x,y,z\rangle$$$ before entering.

### Calculation

Once you've inputted the vectors correctly, click on the "Calculate" button. The calculator will process the input and generate the result.

### Result

The calculated dot product of the two vectors will be displayed in the results section. You can use this information as needed, and if you wish to perform additional calculations, simply clear the existing input by clicking on the "Clear" button, and then enter new vectors.

## What Is the Dot Product Formula?

The dot product, also known as the scalar product or inner product, of two vectors is computed using one of two formulas depending on whether you have Cartesian coordinates or the magnitude and angle between the vectors.

For two vectors in Cartesian coordinates, $$$\mathbf{\vec{u}}=\langle u_1,u_2,u_3\rangle$$$ and $$$\mathbf{\vec{v}}=\langle v_1,v_2,v_3\rangle$$$, the dot product is calculated as:

$$\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}=u_1v_1+u_2v_2+u_3v_3$$If you know the magnitudes of the vectors $$$\mathbf{\vec{u}}$$$ and $$$\mathbf{\vec{v}}$$$, and the angle $$$\phi$$$ between them, the dot product is calculated as:

$$\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}=\mathbf{\left\lvert\vec{u}\right\rvert}\mathbf{\left\lvert\vec{v}\right\rvert}\cos\left(\phi\right)$$In both formulas, $$$\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}$$$ represents the dot product of the vectors $$$\mathbf{\vec{u}}$$$ and $$$\mathbf{\vec{v}}$$$.

## Graphical Interpretation of the Dot Product Between Two Vectors

The dot product of two vectors has an important graphical interpretation. It reflects the geometric relationship between the vectors, specifically, the projection of one vector onto another.

Consider two vectors $$$\mathbf{\vec{u}}$$$ and $$$\mathbf{\vec{v}}$$$ in a two-dimensional or three-dimensional space. When these vectors are drawn from the same starting point (origin), they form an angle $$$\phi$$$ between them.

Projection of One Vector onto Another

The dot product of $$$\mathbf{\vec{u}}$$$ and $$$\mathbf{\vec{v}}$$$ can be interpreted as the length of the projection of $$$\mathbf{\vec{u}}$$$ onto $$$\mathbf{\vec{v}}$$$ multiplied by the magnitude of $$$\mathbf{\vec{v}}$$$ or as the length of the projection of $$$\mathbf{\vec{v}}$$$ onto $$$\mathbf{\vec{u}}$$$ multiplied by the magnitude of $$$\mathbf{\vec{u}}$$$.

This projection can be visualized by dropping a perpendicular from the tip of one vector to the line (or plane in 3D) containing the other vector. The length of this projection represents the coordinate of one vector in the direction of the other.

In mathematical terms:

$$\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}=\mathbf{\left\lvert\vec{u}\right\rvert}\mathbf{\left\lvert\vec{v}\right\rvert}\cos\left(\phi\right)=\mathbf{\left\lvert\vec{v}\right\rvert}\operatorname{scalarproj}_{\mathbf{\vec{v}}}\left(\mathbf{\vec{u}}\right)=\mathbf{\left\lvert\vec{u}\right\rvert}\operatorname{scalarproj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right)$$Angle Between Vectors

The cosine of the angle between two vectors can also be obtained from the dot product:

$$\cos\left(\phi\right)=\frac{\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}}{\mathbf{\left\lvert\vec{u}\right\rvert}\mathbf{\left\lvert\vec{v}\right\rvert}},$$where $$$\phi$$$ is the angle between $$$\mathbf{\vec{u}}$$$ and $$$\mathbf{\vec{v}}$$$, $$$\mathbf{\left\lvert\vec{u}\right\rvert}$$$ and $$$\mathbf{\left\lvert\vec{v}\right\rvert}$$$ are the magnitudes (lengths) of the vectors $$$\mathbf{\vec{u}}$$$ and $$$\mathbf{\vec{v}}$$$, and $$$\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}$$$ is the dot product.

If the dot product of two non-zero vectors is positive, the angle between them is acute. If it's negative, the angle is obtuse. If the dot product is zero, the vectors are perpendicular (or orthogonal) to each other, as $$$\cos\left(90^0\right)=0$$$.

So, the dot product offers a powerful way to understand the geometric relationship between two vectors.

## Why Choose Our Dot Product Calculator?

### Accuracy and Speed

It provides highly accurate results in a matter of seconds, streamlining your calculations and saving valuable time.

### User-Friendly Interface

With an intuitive interface, it's easy to input your vectors and obtain results, whether you're a student, teacher, or professional.

### Versatility

It's capable of handling a wide range of vector dimensions, making it a versatile tool for various mathematical and physical applications.

### Educational Tool

It helps users understand the process of dot product computation, making it a valuable learning resource.

### FAQ

#### What's the difference between a dot product and a cross product?

The dot product results in a scalar quantity and is a measure of the extent to which two vectors are in the same direction. On the other hand, the cross product results in a vector that is orthogonal to the two original vectors, and its magnitude is equal to the area of the parallelogram formed by the two vectors.

#### What if my vectors are denoted in $$$\mathbf{\vec{i}}$$$, $$$\mathbf{\vec{j}}$$$, $$$\mathbf{\vec{k}}$$$ format?

If your vectors are in the $$$\mathbf{\vec{i}}$$$, $$$\mathbf{\vec{j}}$$$, $$$\mathbf{\vec{k}}$$$ format, you can enter them in that format or convert them into the coordinate form before entering them into the calculator.

#### Can I calculate the dot product of vectors in 2D or 3D with this calculator?

Yes, our Dot Product Calculator is versatile and can handle vectors in both 2D and 3D. You just need to enter the vectors' coordinates correctly.

#### How is the dot product calculated?

The dot product of two vectors can be calculated either using their Cartesian coordinates with the formula $$$\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}=u_1v_1+u_2v_2+u_3v_3$$$ or using the magnitudes of the vectors and the angle between them with the formula $$$\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}=\mathbf{\left\lvert\vec{u}\right\rvert}\mathbf{\left\lvert\vec{v}\right\rvert}\cos\left(\phi\right)$$$.