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**chengbin** Given that three vectors $\displaystyle \vec a, \vec b, \vec c$, are perpendicular to each other (where $\displaystyle \vec a, \vec b, \vec c$ are not $\displaystyle \vec 0$ and given that $\displaystyle p\vec a+q\vec b+r\vec c=\vec 0$, show that p=q=r=0.

I get the solution, but I don't understand something written in my solution book.

It says

Since $\displaystyle p\vec a+q\vec b+r\vec c=\vec 0$ <<<<< the result is the null **vector** which is perpendicular to any other vector

$\displaystyle \vec a(p\vec a+q\vec b+r\vec c)=0$ <<<<<< since the null vector is perpendicular to any other vector the scalar (dot) product must be zero. (A dot product yields a real number not a vector!)

Notice that in the second line it is just 0, not $\displaystyle \vec 0$

Why?