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Can somebody please give me an easy application of vector projection in 2D. I dont want to see an example in physics.


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The shadow of a thin object in the plane of the object and the light source.

Component of load force along one member in a joint of two straight beams. (Engineering.)

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I know this, but I am thinking of an application in real world. – Reader Jan 8 '13 at 19:55
Light and shadows aren't the real world? Okay, I'll add another one. – Neal Jan 8 '13 at 20:03

I know it's not specific to $\mathbb{R}^2$, but least squares approximation is a very well-known application of orthogonal projection to very "real world" purposes.

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Think about simple linear regression (OLS) in statistics. In this case, we want to model data $y$ in terms of $x$, such that $$y = \alpha + \beta x + \epsilon.$$

If $y$ is your $n \times 1$ vector of data, then you are projecting it onto two vectors:

  1. An $n \times 1$ vector of one's.
  2. An $n \times 1$ vector of the independent variable $x$.

Bringing these two vectors together as columns of a $n \times 2$ matrix $X$, we can then write the OLS esitmate of $(\alpha, \beta)$ as a parameter vector $\hat{\beta} = (XX')^{-1}X'y$, such that $$E[y] = X\hat{\beta},$$ and the projection matrix is $P = X(XX')^{-1}X',$ thus the expectation is really $$E[y] = Py,$$ or the projection of $y$ onto $x$ and a scalar dimension.

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