Problem on finding a perpendicular vector
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Find a vector perpendicular to $\langle1, 2, 3 \rangle$.
Solution
We recall that
Cross product gives a perpendicular vector
The cross product of two vectors is a vector perpendicular to both.
Finding a perpendicular vector
One way to find a vector perpendicular to $\mathbf{x}$ is to take the cross product with any vector that is not a multiple of $\mathbf{x}$.
For this problem, we will take the cross product of $\langle 1, 2, 3 \rangle$ with $\langle 0, 0, 1\rangle$. The result will be perpendicular to both of these vectors.
Cross product definition
That cross product is $$\langle 1,2,3\rangle \times \langle 0,0,1 \rangle = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 2 & 3 \\ 0 & 0 & 1 \end{vmatrix} = 2 \ \mathbf{i} - \mathbf{j}.$$
Thus, a perpendicular vector to $\langle 1, 2, 3 \rangle$ is $\langle 2, -1, 0 \rangle.$
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