# Thread: Can a right tetrahedron represent a vector?

1. A tetrahedron encloses a region of space with four plane surfaces (triangles). The tetrahedron has six edges. Each edge marks the intersection of two surfaces. It also has four vertices (corners), each vertex marks the intersection of three surfaces.

A “right tetrahedron” may be constructed from four right triangles, this is not a tri-rectangular tetrahedron. A tri-rectangular tetrahedron has three right triangles and one Isosceles triangle.

The four surfaces are all right triangles. A right triangle is shown as a group of three edges in brackets. The longest edge is shown last in each triangle bracket;

(x1, x2, x5) (x5, x3, x4) (x2, x3, x6) (x1, x6, x4)

x12 + x22 = x52 , x52 + x32 = x42 , x22 + x32 = x62 , x12 + x62 = x42

A “component angle” (A1, A2, A3, A4) may be associated with each surface.

x1 = x5Cos(A1) and x2 = x5Sin(A1)
x5 = x4Cos(A2) and x3 = x4Sin(A2)
x2 = x6Cos(A3) and x3 = x6Sin(A3)
x1 = x4Cos(A4) and x6 = x4Sin(A4)

A right tetrahedron may also be associated with a vector (X) in 3D; X = x1e1 + x2e2 + x3e3

Where;
e
1 , e2 , e3 are directional vectors in 3D (basis vectors)
x1 , x2 , x3 are scalar components

The vector has a magnitude; |X| = x4

The scalar components are related to magnitude; x12 + x22 + x32 = x42

Sub-component (x5) is; x52 = x12 + x22 = x42 - x32  2.

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