## Introduction to Orthogonal Projection Calculator:

Orthogonal projection calculator is an online tool that helps you to find the projection of a given vector function. It is used to **calculate vector projection** by multiplying the magnitude with the cosecant of the angle between the two vectors in that direction.

When you find the projection of a vector through manual calculation you may get difficulty in getting the required solution from the vector function. To avoid all this mess try our vector projection calculator which will give you a solution quickly.

## What is Vector Projection?

Vector Projection is used to find the **projection between two vectors** in vector algebra. When you take two vectors a and b then projection points out the same direction as the vector b has. Vector projection is a scalar quantity.

It simplifies the process by breaking down the complex function into tiny pieces to be easily manageable as it is a crucial concept of physics, engineering, or graphic designing.

## Vector Projection Formula:

If you have two vectors a and b then the projection of b is equal to the dot problem of a and b and multiply it with b over the magnitude of b in that direction. The **vector projection formula** used by the Orthogonal projection calculator is,

$$ Projection\; of\; vector\; \vec{A}\; on\; vector\; \vec{B} \;=\; \frac{\vec{A}.\vec{B}}{|\vec{B}|^2} \times \vec{B} $$

## How to Calculate Vector Projection?

For the **calculation of the projection** of one vector onto another, the projection calculator vector follows a systematic approach. Here is a detailed guide about how to solve the vector projection of one vector onto another vector that helps you better understand the process.

**Step 1**:

The projection of a vector calculator determines the given two vectors a (a1,a2,a3,..,an) and b(b1,b2,..,bn) for the vector projection solution.

**Step 2**:

Then, the projection vector calculator computes the Dot Product of Vectors a and b such that,

$$ a ⋅ b \;=\; a_1 . b_1 + a_2 . b_2 + a_3 . b_3 +...+ a_n . b_n $$

**Step 3**:

Calculate the dot product of vector b with itself as b . b,

$$ b ⋅ b \;=\; b_1 . b_1 + b_2 . b_2 + b_3 . b_3 +...+ b_n . b_n $$

**Step 4**:

Then, the orthogonal projection onto subspace calculator applies the vector Projection formula,

$$ Projection\; of\; vector\; \vec{A}\; on\; vector\; \vec{B} \;=\; \frac{\vec{A}.\vec{B}}{|\vec{B}|^2} \times \vec{B} $$

**Step 5**:

Lastly, the projection of u onto v calculator adds the value in the vector projection formula to get the solution of vector projection of vector a onto vector b. By following these steps, you can accurately determine the projection of one vector onto another.

## Solved Example of Vector Projection:

The orthogonal projection calculator helps you to solve the vector projection problems easily but it is also important to understand the manual calculation process. So, an **example** is given below,

### Example: find the following projection,

$$ \vec{u} \;=\; \langle 4, 3 \rangle onto \vec{v} \;=\; \langle 2,8 \rangle $$

**Solution**:

The given vectors are,

$$ \vec{u} \;=\; \langle 4,3 \rangle \; \vec{v} \;=\; \langle 2, 8 \rangle $$

The vector projection of u onto v is given as,

$$ proj_{\vec{v}} \vec{u} \;=\; \frac{\vec{u} . \vec{v}}{||\vec{v}||^2} $$

Add the given vector value to find the projection of u onto v.

$$ proj_{\vec{v}} \vec{u} \;=\; \frac{\langle 4,3 \rangle . \langle 2,8 \rangle}{||\langle 2,8 \rangle ||^2} \langle 2,8 \rangle $$

Solve the above expression into a simplified form,

$$ proj_{\vec{v}} \vec{u} \;=\; \frac{4 . 3 + 3 . 8}{\left( \sqrt{2^2 + 8^2} \right)^2} \langle 2,8 \rangle $$

$$ proj_{\vec{v}} \vec{u} \;=\; \frac{32}{\left( \sqrt{4 + 64} \right)^2} \langle 2,8 \rangle $$

$$ proj_{\vec{v}} \vec{u} \;=\; \frac{32}{ 68} \langle 2,8 \rangle $$

$$ proj_{\vec{v}} \vec{u} \;=\; \frac{8}{ 17} \langle 2,8 \rangle $$

The solution of vector projection u onto v vector is,

$$ proj_{\vec{v}} \vec{u} \;=\; \left \langle \frac{16}{17},\; \frac{64}{17} \right \rangle $$

## How to Use Orthogonal Projection Calculator?

The vector projection calculator has a simple design that allows you to calculate the vector projection question. Follow some of our simple steps before using it to find the solution. These steps are:

**Enter the value**of vector A for vector projection in the input field.- Enter the value of second vector B for vector projection in the input field.
- Review your vector value before pressing the calculate button so that you get the exact solution without any mistakes in this projection calculator vector.
- Click on the “Calculate” button to get the solution of the vector projection problem.
- Click on the “Recalculate” button to get a new page for the evaluation of the vector projection questions.
- If you want to check the accuracy of the solution then you should first try out the load example in this projection of a vector calculator.

## Output of Vector Projection Calculator:

The orthogonal projection onto subspace calculator gives you a **solution** to your given two-vector problem when you click on the calculated button. It may include the following:

**Result Option**:

Result option of the projection vector calculator provides you with solutions for vector projection problems.

**Possible Steps**:

Possible steps provide you with solutions to vector projection questions in a stepwise process.

## Uses of Projection Calculator Vector:

The projection of u onto v calculator has multiple useful features that you get when you use it to calculate the vector projection question and get its solution instantly. These useful features help you in getting in-depth knowledge about the vector projection question for more clarity about this method.

- The projection of vectors calculator
**saves the time**and energy that you consume when you do vector projection problem calculations manually. - Our calculator can operate through a computer, laptop, or mobile which makes it a convenient tool for everyone.
- The projection vector calculator has a user-friendly interface so that you can calculate two vectors projection, and its magnitude easily.
- It provides you with the exact results as per your given input value for the vector projection question with a solution.
- Vector projection calculator is an educational tool that gives you solutions in the step-by-step method so that you can easily become familiar with the concept of vector projection.
- Orthogonal projection calculator is a trustworthy tool as it always provides you with an accurate solution of vector projection problems.