How do you find a matrix of the orthogonal projection onto the line L that consists of scalar multiples

how do you find a matrix of the orthogonal projection onto the line L that consists of scalar multiples of the vector [3,1]?

can anyone show me how to do this?

Here's one method.

Let's suppose you have a vector X, and you want to find the orthogonal projection of some other vector Y onto the line through the origin defined by X.

Well, if X were a unit vector this would be easy. As you probably know, the scalar (dot) product of Y with a unit vector is equal to the magnitude of Y times the cosine of the angle between Y and that vector, which by simple trigonometry is the magnitude of the orthogonal projection that you're looking for. Just multiply that magnitude by the unit vector, and you are done.

But we've got X, so we have to form a unit vector by using (X/|X|), where |X| is the magnitude of X and I'm assuming that X is nonzero. Therefore, your answer is:

The unit vector (X/|X|) multiplied by the scalar product of (X/|X|) with Y.

To write this as a matrix, note that the scalar product of two column vectors A and B can be written as AᵀB (where Aᵀ is the transpose of A). So, if X is a column vector we've got:

orth. proj. of Y = (X/|X|)(X/|X|)ᵀY = (XXᵀ/|X|²)Y = (XXᵀ/(XᵀX))Y

So, the matrix you seek is just (XXᵀ/(XᵀX)).

In your example, X is the column vector:

⎛3⎞
⎝1⎠

so that XᵀX = 3² + 1² = 10. The matrix XXᵀ is found by evaluating

⎛3⎞(3 1)
⎝1⎠

and so XXᵀ/(XᵀX) is, if I haven't made an arithmetic error,

⎛0.9 0.3⎞
⎝0.3 0.1⎠

http://www.wolframalpha.com/ Go there you can type in any math problem, or anything you want to know about and it will give you answer INSTENTLY