Difference between revisions of "The Wires"
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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: | 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: | ||
| − | <center><math>x'=y'\ tan 6^{\circ}+x_0</math></center> | + | <center><math>x'=y'\ tan\ 6^{\circ}+x_0</math></center> |
where <math>x_0</math> is the point where the line crosses the x axis. | where <math>x_0</math> is the point where the line crosses the x axis. | ||
| − | <center><math>y' \Rightarrow {y\ tan 6^{\circ}+x_0, y, 0}</math></center> | + | <center><math>y' \Rightarrow {y\ tan\ 6^{\circ}+x_0, y, 0}</math></center> |
In this form we can easily see that the components of x and y , in the y'-x' plane are | In this form we can easily see that the components of x and y , in the y'-x' plane are | ||
| − | <center><math>x' = y\ sin 6^{\circ}+x_0</math></center> | + | <center><math>x' = y\ sin\ 6^{\circ}+x_0</math></center> |
| − | <center><math>y' = y\ cos 6^{\circ}</math></center> | + | <center><math>y' = y\ cos\ 6^{\circ}</math></center> |
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, | 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, | ||
Revision as of 02:59, 28 April 2017
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:
where is the point where the line crosses the x axis.
In this form we can easily see that the components of x and y , in the y'-x' plane are
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
)