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I had this question: $V$ is Inner product space above $\mathbb C$ and $\langle\cdot,\cdot\rangle$ is the standard inner product. Given $a,b,c \in $ V, and $\|a\|=\|b\|=\|c\|=1$, and $a+b+c=0$. Find $\langle a,b\rangle+\langle b,c\rangle+\langle c,a\rangle$.
I managed to get \begin{align} 0&=\langle a+b+c,a+b+c\rangle \\ &=3+\langle a,b\rangle+\langle b,a\rangle+\langle b,c\rangle+\langle c,b\rangle +\langle a,c\rangle+\langle c,a\rangle \end{align} and got stuck. Thank you.

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    Have you heard about the property called "symmetry" in regards to inner product spaces? – 2017-01-17
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    I know that $ =\overline {} $ – 2017-01-17
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    If we write $x=++$, then collecting terms and their conjugates gives $0=3+x+\overline{x}=3+2\text{Re}(x)$, so we know the real part. Can you find the complex part? – 2017-01-17
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    Not sure.. $x-\overline x$ ? – 2017-01-17
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    Try doing the same thing with $\langle a+b+c, i(a+b+c)\rangle$. – 2017-01-17
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    I tried..could'nt find the complex part.. – 2017-01-17

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