In depth matrices

I need some help understanding matrices in depth. I understand fully how matrices work (how they are added, multiplied, etc). But what I don't quite understand is the underlying relationship in translation matrices. The kind of stuff that the great mathematicians would have spent days trying to figure out. For example:

[cosθ sinθ]
[-sinθ cosθ]

Click to expand...

is the matrix that would rotate a 2-dimensional object clockwise with a magnitude of θ. But what is the relationship here that cause the object to be rotated? I know you simply multiply it by the x and y coordinates. But the main question is, what is the underlying relationship that cause this to work?

Ashford,

I would call that transformation a rotation, not a translation. Basically it involves finding the transformation equations by geometric means, and realizing that the equation's coefficients can be representated by a square matrix. The following link gives you all the "in depth" you would like including three dimensional rotation. Enjoy.

Ratch

Rotation matrix - Wikipedia, the free encyclopedia

Think of all the vectors that comprise an n-dimensional unit sphere.

Multiplying the matrix by all those unit vectors generates another n-dimensional ellipse/plane/line/point obtained by rotating and/or stretching the original sphere.

But what is the relationship here that cause the object to be rotated?

Click to expand...

Perhaps your question is, "What is the application of matices to physics?"

The answer is that it comes from the problem statement: initial conditions, forces, torques, etc.

Examples:

1. What is the path traced by a point on the tread of a tire on a car traveling at 30 mph? The diameter of the tire is 24in.

2. Consider a playground merry-go-round:

Sport Play 301-146G 10' Merry Go Round in Tan and Green

The merry-go-round is set in motion at .25 revolutions/sec. A child on the rim walks toward the center at 1 ft/sec. What is the child's acceleration? The child weighs 50 lb. What forces does he feel?

When you have two dimensional coordinates, say (x, y), you can plot that point using some basis vectors. Usually the coordinate point is plotted using the x and y axes of the cartesian coordinate system (the basis vectors being i and j).

You can also define your own basis vectors. If you look at the matrix:

[ cosθ sinθ]
[-sinθ cosθ]

You notice that the column vectors are unit vectors (their length is one) and that the vectors are also orthogonal. When you multiply this matrix by a vector containing your coordinates x and y:

[ cosθ sinθ] [x]
[-sinθ cosθ] [y]

You notice that the x coordinate scales the first column vector and the y coordinate scales the second column vector.. and finally the two vectors are added.

So, the matrix basically defines new basis vectors to represent your coordinate system (relative to some "world" coordinate system).
Draw the column vectors [cosθ -sinθ]' and [sinθ cosθ]' on a paper (with some fixed angle θ) and you will easily see how the system works.

MisterT that was exactly what I wanted to know. Thanks