- #1
yungman
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This is from "Field and Wave Electromagnetics" by Cheng. I don't understand how this work:
Let
\vec k \;=\; \hat x k_x + \hat y k_y + \hat z k_z \;\hbox { and }\; \vec R \;=\; \hat x x + \hat y y + \hat z z
be position vectors.
Find the equation of the plane perpendicular to \vec k and contain the tip of vector \vec k
ie: plane contain point P(k_x, k_y, k_z)
Let k^2 = k_x ^2 + k_y ^2 + k_z ^2.
The book said
\hat k \cdot \vec R = \hbox { constant }\;. is the equation of the plane!
Using the book’s formula and call the constant as A:
\hat k \cdot \vec R \;=\; \frac { \hat x k_x + \hat y k_y + \hat z k_z }{ k} \cdot (\hat x x + \hat y y + \hat z z) = \frac { x k_x + y k_y + z k_z }{k} = A
\Rightarrow \; x k_x + y k_y + z k_z \;=\; kA (1) is equation of plane.
Normal way of finding equation of plane using point normal is
\vec k \cdot (\vec R - \vec k )=0
\vec k \cdot (\vec R - \vec k ) \;=\; (\hat x k_x + \hat y k_y + \hat z k_z ) \cdot [(\hat x x + \hat y y + \hat z z)\;-\; (\hat x k_x + \hat y k_y + \hat z k_z )]
\Rightarrow\; x k_x + y k_y + z k_z = k^2 (2)
As you can see (1) is not the same as (2)
Can anyone explain to me?
Let
\vec k \;=\; \hat x k_x + \hat y k_y + \hat z k_z \;\hbox { and }\; \vec R \;=\; \hat x x + \hat y y + \hat z z
be position vectors.
Find the equation of the plane perpendicular to \vec k and contain the tip of vector \vec k
ie: plane contain point P(k_x, k_y, k_z)
Let k^2 = k_x ^2 + k_y ^2 + k_z ^2.
The book said
\hat k \cdot \vec R = \hbox { constant }\;. is the equation of the plane!
Using the book’s formula and call the constant as A:
\hat k \cdot \vec R \;=\; \frac { \hat x k_x + \hat y k_y + \hat z k_z }{ k} \cdot (\hat x x + \hat y y + \hat z z) = \frac { x k_x + y k_y + z k_z }{k} = A
\Rightarrow \; x k_x + y k_y + z k_z \;=\; kA (1) is equation of plane.
Normal way of finding equation of plane using point normal is
\vec k \cdot (\vec R - \vec k )=0
\vec k \cdot (\vec R - \vec k ) \;=\; (\hat x k_x + \hat y k_y + \hat z k_z ) \cdot [(\hat x x + \hat y y + \hat z z)\;-\; (\hat x k_x + \hat y k_y + \hat z k_z )]
\Rightarrow\; x k_x + y k_y + z k_z = k^2 (2)
As you can see (1) is not the same as (2)
Can anyone explain to me?
