**FUNDAMENTAL PROBLEMS IN GEOMETRY**

**Trigonometry**

The trigonometric functions of a general angle: Let 0 be an angle with
its vertex at the origin and its initial side coincident with the x-axis, and
let (x, y) be any point in the Cartesian plane distinct from the origin, and on
the terminal side of the angle. The six trigonometric functions are as follows:

**Polar Coordinates**

We are used to the standard Cartesian coordinate system of graphing, but
there are many other methods. One common and very useful method is the polar
coordinate method. In this system, the origin is at the center of the plane and
one specifies coordinates by giving an angle 0 and a radius r:

If you need to translate polar coordinates to Cartesian coordinates, use
these formulas:

**3-D Coordinate Systems**

In a 3-dimensional Cartesian system, we specify 3 values to identify a
point (x, y, z):

This is a right-handed system which is most often used in graphics,
i.e., the thumb of the right hand points down the z-axis if we imagine grabbing
the z-axis with the fingers of the right hand curling from the positive x-axis
toward the positive y-axis.

**Matrices**

A matrix is a rectangular array of quantities, called
the elements of a matrix. We identify matrices according to the number of rows
X number of columns.

When number of rows = number of columns, we have a square matrix. A matrix with a single
row or column represents a vector, and a larger matrix can be viewed as a
collection of row or column vectors.

To multiply a matrix

**A**by a scalar value s, we just multiply each element of the matrix by s. Matrix addition is defined only for matrices with the same number of rows and columns and is simply the sum of corresponding elements.
We can multiply an m X n matrix A by a p X q matrix B only if n = p. We
obtain an m X q matrix

**C**whose elements are calculated as in the following example:
Vector multiplication in matrix notation produces the same result as the
inner product providing the first vector is a row vector and the second is a
column vector. The transpose of
a matrix AT is obtained by interchanging rows and columns.

The matrix B is the inverse of
a matrix A if AB = I where I is the identity matrix, being all O's except for
1's on the diagonal.

## No comments:

## Post a Comment