The Wires

We can parametrize the equations for the wires and wire midpoints to express the equation in vector form. In the y'-x' plane the general equation follows the relationship:

[math]x'=y' tan 6^{\circ}+x_0[/math]

where [math]x_0[/math] is the point where the line crosses the x axis.

[math]y' \Rightarrow {y tan 6^{\circ}+x_0, y, 0}[/math]

In this form we can easily see that the components of x and y , in the y'-x' plane are

[math]x' = y sin 6^{\circ}+x_0[/math] [math]y' = y cos 6^{\circ}[/math]

The parameterization has reduced two equations with two variables, to two equations which depend on one variable. Working in the y-x plane, we will undergo a positive rotation,

R(Subscript[\[Theta], yx])=(cos 6\[Degree] -sin 6\[Degree] 0 sin 6\[Degree] cos 6\[Degree] 0 0 0 1

)

(Components of same vector in new system

)=(Passive transformation matrix

) . (Components of vector in original system

) (New basis vectors

)=(Active transformation matrix

) . (original basis vectors

)

(x y z

)=(cos 6\[Degree] -sin 6\[Degree] 0 sin 6\[Degree] cos 6\[Degree] 0 0 0 1

) . (x' y' z'

) (x' y' z'

)=(cos 6\[Degree] sin 6\[Degree] 0 -sin 6\[Degree] cos 6 \[Degree] 0 0 0 1

) . (x y z

)

(x y z

)=(cos 6\[Degree] -sin 6\[Degree] 0 sin 6\[Degree] cos 6\[Degree] 0 0 0 1

) . ( y sin 6\[Degree]+Subscript[x, 0] y cos 6\[Degree] 0

)

(x y z

)= (-y cos 6 \[Degree] sin 6 \[Degree]+Subscript[x, 0]cos 6 \[Degree] +y cos 6 \[Degree]sin 6 \[Degree] y cos^2 6 \[Degree]+Subscript[x, 0]sin 6 \[Degree]+y sin^2 6 \[Degree] 0

) (x' y' z'

)= (x cos 6\[Degree]+y " sin 6\[Degree] -x sin 6 \[Degree]+y " cos 6\[Degree] 0

)

(x y z

)= (Subscript[x, 0]cos 6 \[Degree] y +Subscript[x, 0]sin 6 \[Degree] 0

)