A complex number $z = a + bi$ represents a vector $\langle a,b\rangle$.
How to distinguish complex multiplication ($z_1 \cdot z_2$) from dot product ($z_1\cdot z_2 = a_1a_2+b_1b_2$)?
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$\begingroup$ Context$~~\!\!\!$ $\endgroup$– JMoravitzCommented Mar 30, 2022 at 16:19
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$\begingroup$ @JMoravitz Say you want to prove dot product is invariant under rotation: $(A\cdot v)\cdot (B\cdot v)$. $\endgroup$– AvidSeekerCommented Mar 30, 2022 at 16:35
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1$\begingroup$ A two dimensional vector if referred to as a two dimensional vector will get the dot product if no further context is provided. You would never use complex multiplication for things which you do not have reason to believe are complex numbers. If you are talking about complex numbers and name them as such, then when multiplying two complex numbers you do so using complex multiplication. You would never say "let $z,w\in\Bbb C$. Consider $z\cdot w$" and have the multiplication be meant to be dot product. $\endgroup$– JMoravitzCommented Mar 30, 2022 at 16:52
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$\begingroup$ Similarly... if you have a scalar $\alpha$ and a vector $v$, when you do $\alpha \cdot v$ this is the scalar product... not a dot product and not a complex multiplication... or if you have two scalars $\alpha \cdot \beta$ this is the multiplication of the scalar field... You very simply use the multiplication appropriate for the setting you are in. There is rarely if ever any confusion in the matter. It is like being confused whether we are talking about a karate uniform (Gi) or indian butter (Ghee). If we're in a dojo, its the uniform. If we're in a kitchen, its the butter. $\endgroup$– JMoravitzCommented Mar 30, 2022 at 16:55
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$\begingroup$ IF you happen to be in a situation where it does become ambiguous and either could be meant... then make it unambiguous by being more specific what you are referring to. Use different symbols. You can use $\boxdot$ or $\otimes$ or any number of other less commonly used symbols, or use subscripts like $\times_\Bbb C$ and so on... its not hard. $\endgroup$– JMoravitzCommented Mar 30, 2022 at 16:58
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2 Answers
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The complex product of two complex numbers $z_0$ and $z_1$ is simply denoted as $z_0\cdot{z_1}$ or $z_0z_1.$ Meanwhile, the standard inner product is $\langle{z_0,z_1}\rangle=a_0a_1+b_0b_1=\mathrm{Re}[(a_0a_1+b_0b_1)+(a_1b_0-a_0b_1)i]=\mathrm{Re}(z_0z_1^*),$ where $\mathrm{Re}$ denotes the real part function, and $^*$ denotes the complex conjugate. That is what distinguishes the two. There is no situation where you would use the notation $z_0\cdot{z_1}$ for the inner product, because whenever you use $\cdot$ for the complex product, you can write the inner product as $\mathrm{Re}(z_0\cdot{z_1^*})=\mathrm{Re}(z_0^*\cdot{z_1}).$
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Using angle brackets for a vector is very uncommon. It is common however to denote scalar products: $$\langle a, b\rangle := a_1b_1 + a_2b_2$$ The "ordinary" product of complex numbers $a$ and $b$ is denoted as $ab$ or as $a\cdot b$.
answered Mar 30, 2022 at 16:30