Welcome to today’s blog post where we will be discussing the orthogonal projection formula. The orthogonal projection is a fundamental concept in linear algebra that allows us to find the closest point in a subspace to a given vector. Understanding this formula is crucial in various fields such as computer graphics, physics, and engineering.

### Definition

The orthogonal projection of a vector *v* onto a subspace *U* is the vector *p* which is the closest point in *U* to *v*. This projection vector *p* is obtained by finding the linear combination of the basis vectors of *U* that minimizes the distance from *v*.

### Formula

The orthogonal projection formula can be expressed as:

*p* = *u* – (*u* · *n*) * *n*

Where:

*p*is the orthogonal projection vector*u*is the vector being projected*n*is the unit normal vector to the subspace*U**·*represents the dot product operation

### Example

Let’s consider a simple example to illustrate the orthogonal projection formula. Suppose we have a vector *v* = (3, 4) and a subspace *U* defined by the equation *x* – *y* = 0. To find the orthogonal projection of *v* onto *U*, we first need to determine the unit normal vector *n*. In this case, *n* = (1, -1). Plugging the values into the formula, we get:

*p* = (3, 4) – ((3, 4) · (1, -1)) * (1, -1)

*p* = (3, 4) – (3 – 4) * (1, -1)

*p* = (3, 4) – (-1) * (1, -1)

*p* = (3, 4) + (1, -1)

*p* = (4, 3)

Therefore, (4, 3) is the orthogonal projection of (3, 4) onto the subspace *U*.

### Applications

The orthogonal projection formula has various applications in different fields:

- In computer graphics, it is used to create realistic shadows and reflections.
- In physics, it is applied to calculate the work done by a force in a particular direction.
- In engineering, it helps in determining the components of a force acting on a structure.

### Limitations

It’s important to note that the orthogonal projection formula assumes that the subspace *U* is a closed and finite-dimensional subspace. If *U* is not finite-dimensional, the formula may not apply directly, and alternative methods need to be used.

## Conclusion

The orthogonal projection formula is a powerful tool in linear algebra that allows us to find the closest point in a subspace to a given vector. Understanding this formula is essential in many areas of science and engineering. I hope this blog post has provided a clear explanation of the orthogonal projection formula. If you have any questions or additional insights, please feel free to leave a comment below.

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