# Vector Projection Formula

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 vector projection of one vector over another vector is the length of the shadow of the given vector over another vector. 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, and this on further simplification gives the final vector projection formula.

## What is Vector Projection Formula?

Projection of vector \(\vec A\) over vector \(\vec B \) is obtained by product of vector A with the Cosecant of the angle between the vectors A and B. The vector projection can be quickly calculated by the below formula.

### \(\text{Projection of Vector} \vec {A} \ \text{on Vector} \vec{B} = \dfrac{\vec{A}. \vec{B}}{| \vec{B}|}\)

**Break down tough concepts through simple visuals.**

Here let us check a few solved examples to understand how to use the vector projection formula.

## Solved Examples on Vector Projection Formula

### Example 1:** **Find the projection of the vector \(4\hat i + 2\hat j + \hat k\) on the vector \(5\hat i -3\hat j + 3\hat k\).

**Solution:**

Given \(\vec A = 4\hat i + 2\hat j + \hat k\) and \(\vec B = 5\hat i -3\hat j + 3\hat k\).

\(\begin{align}\text{Projection of Vector} \vec {A} \ \text{on Vector} \vec{B} = \dfrac{\vec{A}. \vec{B}}{| \vec{B}|}\\&=\dfrac{(4.(5) + 2(-3) + 1.(3))}{|\sqrt{5^2 + (-3)^2 + 3^2}|}\\&=\dfrac{17}{\sqrt{43}}\end{align}\)

### Example 2:** **Find the projection of the vector \(5\hat i + 4\hat j + \hat k\) in the direction of the vector \(3\hat i + 5\hat j -2\hat k\).

**Solution:**

Given \(\vec A = 5\hat i + 4\hat j + \hat k\) and \(\vec B = 5\hat i -3\hat j + 3\hat k\).

\(\begin{align}\text{Projection of Vector} \vec {A} \ \text{on Vector} \vec{B} = \dfrac{\vec{A}. \vec{B}}{| \vec{B}|}\\&=\dfrac{(5.(3) + 4(5) + 1.(-2))}{|\sqrt{3^2 + 5^2 + (-2)^2}|}\\&=\dfrac{33}{\sqrt{38}}\end{align}\)