# If a vector n is orthogonal to vectors a and b, is it also orthogonal to any linear combination of a and b?

It makes sense conceptually to me I would just like this verified.

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The dot product can be distributed over addition: $\mathbf n\cdot(p\mathbf a+q\mathbf b)=p(\mathbf n\cdot\mathbf a)+q(\mathbf n\cdot\mathbf b)$... –  Ｊ. Ｍ. Oct 18 '11 at 1:12
Yes! One way to check that two vectors are orthogonal is that their dot product is zero. So now you just need to algebraically check that if $\vec{v}\cdot\vec{a}=0$ and $\vec{v}\cdot\vec{b}=0$ then for any scalars $r,s$ we have $$\vec{v}\cdot(r\vec{a}+s\vec{b})=0.$$