Sum of the dot product of complex vectors

[happyparticle]

Summary:: summation of the components of a complex vector

Hi,

In my textbook I have
$\widetilde{\vec{E_t}} = (\widetilde{\vec{E_i}} \cdot \hat{e_p}) \hat{e_p}$

$\widetilde{\vec{E_t}} = \sum_j( (\widetilde{\vec{E_{ij}}} \cdot {e_{p_j}}*) \hat{e_p}$

For $\hat{e_p} = \hat{x}$
$\widetilde{\vec{E_{0i}}} = (E_{0x} \hat{x} \pm E_{0y} \hat{y} )e^{i\theta}$

Thus, $\widetilde{\vec{E_0t}} = (\sum E_{0ij} e_{pj}*) \hat{e_p}$

and then
$\widetilde{\vec{E_0t}} = E_0 cos \theta \hat{x} + E_0 sin^* \theta (0)$

I don't see how to get the last line. This is what I think.

$\widetilde{\vec{E_0t}} = [(E_{0x}e^{i \theta}] \hat{x}$ where, $e_{p}* = x - iy = 1$ because $\hat{e_p} = \hat{x}$

= $E_{0x} (cos \theta + i sin \theta) \hat{x}$

I tried different way to get the same result, but I'm not sure to fully understand.

 

[Philip Koeck]

Summary:: summation of the components of a complex vector

Hi,

In my textbook I have
$\widetilde{\vec{E_t}} = (\widetilde{\vec{E_i}} \cdot \hat{e_p}) \hat{e_p}$

$\widetilde{\vec{E_t}} = \sum_j( (\widetilde{\vec{E_{ij}}} \cdot {e_{p_j}}*) \hat{e_p}$

For $\hat{e_p} = \hat{x}$
$\widetilde{\vec{E_{0i}}} = (E_{0x} \hat{x} \pm E_{0y} \hat{y} )e^{i\theta}$

Thus, $\widetilde{\vec{E_0t}} = (\sum E_{0ij} e_{pj}*) \hat{e_p}$

and then
$\widetilde{\vec{E_0t}} = E_0 cos \theta \hat{x} + E_0 sin^* \theta (0)$

I don't see how to get the last line. This is what I think.

$\widetilde{\vec{E_0t}} = [(E_{0x}e^{i \theta}] \hat{x}$ where, $e_{p}* = x - iy = 1$ because $\hat{e_p} = \hat{x}$

= $E_{0x} (cos \theta + i sin \theta) \hat{x}$

I tried different way to get the same result, but I'm not sure to fully understand.

Shouldn't e_p* be a vector?
You write e_p* = x -iy. That's a scalar.

 

[Paul Colby]

Too many undefined notational symbols to guess at, which textbook are you using?