I Sum of the dot product of complex vectors

AI Thread Summary
The discussion centers on the summation of components in complex vectors, specifically how to derive the expression for the total electric field vector, ##\widetilde{\vec{E_t}}##, from its components. The user is attempting to understand the relationship between the complex vector representation and its projection onto a specific direction, ##\hat{e_p}##. There is confusion regarding the notation used, particularly the definition of ##e_{p}*##, which is questioned as being a scalar rather than a vector. The user expresses uncertainty about reaching the final expression and seeks clarification on the notation and derivation process. Overall, the thread highlights the complexities of working with complex vectors in physics and the need for clear definitions in mathematical expressions.
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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.
 
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EpselonZero said:
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 ep* be a vector?
You write ep* = x -iy. That's a scalar.
 
Too many undefined notational symbols to guess at, which textbook are you using?
 
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