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# Vector Projection Calculator

Vectors are quantities with both magnitude and direction. Vectors help to simultaneously represent different quantities in the same expression. The vector projection of one vector over another is obtained by multiplying the given vector with the cosecant of the angle between the two vectors.

## What is Vector Projection Calculator?

Vector Projection Calculator' is an online tool that helps to calculate the vector projection for the given two vectors. Online Vector Projection Calculator helps you to calculate the vector projection for the given two vectors in a few seconds.

### Vector Projection Calculator

**NOTE:** Enter the values up to two digits only.

## How to Use Vector Projection Calculator?

Please follow the steps below on how to use the calculator:

**Step 1:**Enter the coefficients of two vectors in the given input boxes.**Step 2:**Click on the**"Calculate"**button to calculate the vector projection for the given two vectors.**Step 3:**Click on the**"Reset"**button to clear the fields and enter the new values.

## How to Find Vector Projection?

The vector projection formula gives the projection of one vector over another vector. The resultant of the vector projection formula is a scalar value.

The standard form of representation of a vector is:

A = a_{1}i^ + b_{1}j^ + c_{1}k^

B = a_{2}i^ + b_{2}j^ + c_{2}k^

Where a_{1}, b_{1}, c_{1}, and a_{2}, b_{2}, c_{2} are numeric values, and i^, j^, k^ are the unit vectors along the x-axis, y-axis, and z-axis respectively.

The formula to calculate the vector projection is given by

\(\text{Projection of Vector} \overrightarrow{A} \ \text{on Vector} \overrightarrow{B} = \dfrac{\overrightarrow{A}. \overrightarrow{B}}{| \overrightarrow{B}|^2} × \overrightarrow{B}\)

Let us see an example to understand briefly.

## Solved Example on Vector Projection Calculator

**Example:**

Find the vector projection for given two vectors a = 4i + 2j - 5k and b = 3i - 2j + k and verify it using the online vector projection calculator.

**Solution:**

Given A = 4i + 2j - 5k and B = 3i - 2j + k

**\(\text{Projection of Vector} \overrightarrow{A} \ \text{on Vector} \overrightarrow{B} = \dfrac{\overrightarrow{A}. \overrightarrow{B}}{| \overrightarrow{B}|^2} × \overrightarrow{B}\)**

\(\overrightarrow{A}. \overrightarrow{B}\) = (4i + 2j – 5k) . (3i - 2j + k)

= (4 × 3) + (2 × -2) + (-5 × 1)

= 12 - 4 - 5

= 3

\(| \overrightarrow{B}|^2\) = \(|\sqrt{3^2 + (-2)^2 + 1^2}|^2\)

\(= {14}\)

\(= \dfrac{\overrightarrow{A}. \overrightarrow{B}}{| \overrightarrow{B}|^2} × \overrightarrow{B}\)

\(= \frac{3}{14} × (3i - 2j + k)\)

= (9i - 6j + 3k) / 14

Therefore , the vector projection for given two vectors is (9i - 6j + 3k) / 14 or (9/14, -6/14, 3/14)

Similarly, you can use the vector projection calculator to find the vector projection for given two vectors:

- a = 4i + 2j - 5k and b = -1i + 4j - 3k
- a = -2i - 5k and b = -7i + j + k

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