Please help explaining equation of the plane from the book.

[yungman]

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?

 

[tiny-tim]

hi yungman!

the plane has to "contain the tip of vector k" …

ie R = k must work …

so the constant A in the equation k.R = A must be equal to k.k, ie k² …

so (1) and (2) are the same, with A = k²

 

[yungman]

Thanks Tiny-Tim.