Urgent! Equations for Planes and Vectors in 3D???

Question

Ok, I am trying to build a pyramid in coordinates (x,y,z). The base has 4pts (10,10) (10,-10) (-10,10) (-10,-10) that make 4lines that make a perfect square. The base lies in the xy plane only, z=0. The pyramid sticks above the xy plane.The peak of the pyramid is (0,0,50). How do I find the equation of the 4 slanted lines and the 4planes (the ones in 3d) that make the pyramid? Need answer today or tommorow. Any help would be appreciated.

Answer 1

I think you mean that the coordinates of the base are
(10, 10, 0), (-10, 10, 0), (-10, -10, 0), and (10, -10, 0). You need to show the z coordinate as well, although you did say it is in the plane z = 0.

The vertex is (0, 0, 50).

A line is defined by two points. The first line goes thru the points (10, 10, 0) and (0, 0, 50).

The directional vector is:
u = <10-0, 10-0, 0-50> = <10, 10, -50>

Any non-zero multiple is also a directional vector of the line. Divide by 10.

u = <1, 1, -5>

Now write the equation of the line with one of the points on the line and the directional vector u. Let's choose (10, 10, 0).

L1 = <10, 10, 0> + t<1, 1, -5>
where t is a scalar ranging over the real numbers

The equations for the other three lines can be similarly derived.
_______________

Now let's find one of the planes.

Let the plane be defined by the two points above, namely (10, 10, 0) and (0, 0, 50) and a third point (-10, 10, 0).

We can get a second directional vector v, that lies in the plane. It is defined by the points (10, 10, 0) and (-10, 10, 0).

v = <10 - -10, 10-10, 0-0> = <20, 0, 0>

Any non-zero multiple is also a directional vector of the line. Divide by 20.

v = <1, 0, 0>

To find the normal vector n, of the plane take the cross product of the two vectors u and v that lie in the plane.

n = u X v = <1, 1, -5> X <1, 0, 0> = <0, -5, -1>

With the normal vector n and one point in the plane we can write the equation of the plane. Let's choose (0, 0, 50).

0(x - 0) - 5(y - 0) - 1(z - 50) = 0
-5y - z + 50 = 0
5y + z - 50 = 0

The other three planes can be determined similarly.