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I am trying to find values for a and b for the following set of vectors:

$\left(\left[\begin{matrix} 1\\2\\3\end{matrix}\right],\left[\begin{matrix} 4\\1\\-2\end{matrix}\right], \left[\begin{matrix} a\\b\\3\end{matrix}\right]\right) = 0$

I know the idea is that, if I take the dot product of all the column vectors in the above vector space, I should end up with zero. But I am not sure how to get started solving for a and b, such that the below set of vectors is orthogonal. A hint as to how to get started?

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    Sorry, you are right. I should have said take the dot product of all pairs of vectors.2012-03-19

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In order for $(1,2,3)^t$ to be orthogonal to $(a,b,3)^t$, we must have $a + 2b + 9 = 0.$ In order for $(4,1,-2)^t$ to be orthogonal to $(a,b,3)^t$, we must have $4a + b - 6 = 0.$ $(1,2,3)^t$ and $(4,1,-2)^t$ are orthogonal, regardless of the values of $a$ and $b$.

So what you need to do is solve the $2\times 2$ system of linear equations $\begin{array}{rccccl} a & + & 2b & = & -9\\ 4a & + & b & = & 6. \end{array}$

Do you know how to solve systems of linear equations? (Gaussian-Jordan elimination, for instance?)

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    Solving the above system I found the following: \left[\begin{matrix}1&0&3\\0&1&-6\end{matrix}\right] . I made sure to check each pairewise dot product, where $a=3$ and $b=-6$, and it does check out. Thanks for the tip.2012-03-19