Questions on how the projection works

[Bio_J]

Hi team,
in the video Dimensionality Reduction and Projection
I don’t get it why two vector multiply together will get the direction on the same line?

Like in the picture, [1.2, 1.6] * [1,1] will get the point on the same direction as [1,1]

" To choose that orange vector note that the line y equals x is actually

the span of the vector with coordinates 1, 1.

So let’s use that vector 1, 1.

Now if you take the dot product of that first row in the table and the orange

vector, you’re essentially saying take 1 times the point x coordinate and

one time the point y coordinate to find the new location projected along the line."

I understand that two vector’s dot product is equal to l2-norm u * l2-norm v * cosin, if norm are on the same direction then cosin = 1, but I can’t relate this to two different direction vector… as the angle is not 0, so where is the cosin?

[TMosh]

This paper seems to be a nice proof:

lec1snip2-dotprod.pdf

[paulinpaloalto]

Sorry to go all math geek here, but what they show is that if you assume that the Law of Cosines is already proven, you can derive our desired formulation from that:

A \cdot B = |A||B|cos(\theta)

So what they’ve shown is that the two statements are equivalent, but they didn’t really prove either of them.

But it turns out it’s not that hard to prove the Law of Cosines by using the definitions of sin, cos and the Pythagorean Theorem from which you can also easily get the familiar identity:

sin^2\theta + cos^2\theta = 1