Hello elementary school students.
We are going to formalize some math and geometry today.
We start with our standard coordinate system in
this, our four dimensional universe.

We can move left or right, we call this our X axis.
We can move up or down, we call this our Y axis.
We can move back or forward, we call this our Z axis.
We can move in or out, we call this our W axis.

You all know how to use your computer for
add +, subtract -, multiply *, divide /, and
compute power x^n pow(x,n),  square root sqrt(x),
sine sin(x), cosine cos(x), arc sine asin(a)
determinant det(m), and many other math functions.

In our coordinate system, we need to know where
something is. We use the term "point" for a
zero dimensional object with coordinate values
x, y, z, w.

We can have one dimensional objects that go from
point A to point B, called a "line". A line has
a length computed as:
L = sqrt((Ax-Bx)^2 + (Ay-By)^2 + (Az-Bz)^2 + (Aw-Bw)^2))
A line has 2 points and 1 line.

We can have two dimensional objects, the simplest being
a square, moving a line parallel to itself, to get
a square. If the line was length L and was moved a
distance L parallel, we have a surface of size L^2 .
A square has 4 points and 4 lines.

We can have three dimensional objects, the simplest being
a cube, moving square parallel to its sides, to get
a cube. If the L by L square was moved a distance L
parallel to its sides, we have an area of size L^3 .
A cube has 8 points, 12 lines, 6 surfaces. 

We can have a four dimensional object, the simplest being
a box, moving a cube parallel to its sides, to get
a box. If the L by L by L cube was moved a distance L
parallel to its sides, we have a box with a volume L^4.
A box has 16 points, 32 lines, 24 surfaces, 8 cubes.

Now we change to geometry where we have a radius R
about some point P.

The two dimensional object is a circle created by
having points at a distance R from point P.
The length of the circle is 2 Pi R and the
surface of the circle is Pi R^2.
Pi approximately 3.1415926535897932 or from math library.
The equations for the points, with angle a1 going
from 0 to 2 Pi in X,Y are:
x = R*cos(a1) + Px
y = R*sin(a1) + Py

Given x,y each -Px,-Py we can compute R and a1:
R = sqrt(x^2 + y^2)
a1 = acos(x/R)

The three dimensional object is a sphere created by
having points at a distance R in two dimensions
from point P. The surface is now 4 Pi R^2
and the area is now 4/3 Pi R^3 .
The equations for the points, with angle a1 in X,Y
angle a2 in Y,Z each going from 0 to 2 Pi:
x = R*cos(a1) + Px
y = R*sin(a1)*cos(a2) + Py
z = R*sin(a1)*sin(a2) + Pz

Given x,y,z each -Px,-Py,-Pz we can compute R and a1, a2:
R = sqrt(x^2 + y^2 + z^2)
a1 = acos(x/R)
a2 = acos(y/sqrt(y^2 + z^2))

The four dimensional object is a ball created by
having points at a distance R in three dimensions
from point P. The area is now 2 Pi^2 R^3 and the
volume of the ball is 1/2 Pi^2 R^4 .
The equations for the points, with angle a1 in X,Y
angle a2 in Y,Z angle a3 in Z,W each going from 0 to 2 Pi:
x = R*cos(a1) + Px
y = R*sin(a1)*cos(a2) + Py
z = R*sin(a1)*sin(a2)*cos(a3) + Pz
w = R*sin(a1)*sin(a2)*sin(a3) + Pw

Given x,y,z,w each -Px,-Py,-Pz,-Pw we can compute R
and a1, a2, a3:
R = sqrt(x^2 + y^2 + z^2 + w^2)
a1 = acos(x/R)
a2 = acos(y/sqrt(y^2 + z^2 + w^2))
a3 = acos(z/sqrt(z^2 + w^2))

We now formalize vector and matrix and the checking
of two vectors being orthogonal. A vector is a specific
line from point A to point B. One definition of a
vector can be [Ax,Ay,Az,Aw][Bx,By,Bz,Bw] eight values.
Given point A, we can use the equations of a ball to
get point B from three angles and a length R using
x = Bx-Ax, y = By-By, z = Bz-Az, w = Bw-Aw.
[Ax,Ay,Az,Aw][R,a1,a2,a3]
The angles define the direction of the vector.
Formally in mathematics, orthogonal vectors have
a dot product of zero.
The cross product can be used to compute a
vector D orthogonal to vectors A, B, C.

The dot product and cross product of vectors are define as:
dot(A,B) = sqrt(Ax*Bx + Ay*By + Az*Bz + Aw*Bw)  a scalar.

D = cross(A,B,C) = [detx, dety, detz, detw]

          |Ay,Az,Aw|            |Ax,Az,Aw|
detx = det|By,Bz,Bw|  dety = det|Bx,Bz,Bw|
          |Cy,Cz,Cw|            |Cx,Cz,Cw|

          |Ax,Ay,Aw|            |Ax,Ay,Az|
detz = det|Bx,By,Bw|  detw = det|Bx,By,Bz|
          |Cx,Cy,Cw|            |Cx,Cy,Cz|

Check dot(A,D)=0  dot(B,D)=0  dot(C,D)=0

When you get into high school, mathematics will cover
calculus. This includes integration and differentiation
with definitions such as: The integral of sin(x) dx is
cos(x) and the derivative of cos(x) is sin(x) and
many many more. All of the mathematics is defined for
four variables and more, derivatives may be 4th order
and more, integration may apply to four or more
variables.

You know length measurement may be in English units
inch, foot, mile etc. And in Metric units millimeter,
meter, kilometer, etc.

The laws of physics, such as force, mass, acceleration
f = m * a, etc. apply in four dimensions. Units change,
such as density typically has units kilogram per meter^4.

When you get to college you will hear the theory that
our four dimensional universe is a subset of a five
dimensional universe. All the mathematics and physics
will be shown to extend to five dimensions.

End of this lecture, lots more to learn.