How to find perpendicular vector to another vector?

Question 1

How do I find a vector perpendicular to a vector like this: $3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$
Could anyone explain this to me, please?

I have a solution to this when I have $3\mathbf{i}+4\mathbf{j}$ , but could not solve if I have $3$ components…

When I googled, I saw the direct solution but did not find a process or method to follow. Kindly let me know the way to do it. Thanks.

Question 2

There exists an infinite number of vectors in 3 dimension that are perpendicular to a fixed one.
They should only satisfy the following formula:
$(3\mathbf{i}+4\mathbf{j}-2\mathbf{k}) \cdot v=0$

For finding all of them, just choose 2 perpendicular vectors, like $v_1=(4\mathbf{i}-3\mathbf{j})$ and $v_2=(2\mathbf{i}+3\mathbf{k})$ and any linear combination of them is also perpendicular to the original vector: $v=((4a+2b)\mathbf{i}-3a\mathbf{j}+3b\mathbf{k}) \hspace{10 mm} a,b \in \mathbb{R}$

How to find perpendicular vector to another vector?

Answer

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