Find all vectors $X=(x,y,z)$ for which $(1,3,5)\cdot X=(2,4,6)\cdot X$.

Question

Find all vectors $X=(x,y,z)$ for which $(1,3,5)\cdot X=(2,4,6)\cdot X$.

This is from an homework assignment I feel like I am right there but just missing the last part.

I did a dot product of the two vectors with $X$ and set them equal to each other.

$$ x+3y+5z = 2x+4y+6z $$ This equals $$-x-y-z = 0$$

Now this is a plane equation and all vectors in this plane fulfill the question above, so doesn't that mean there are an infinite amount of vectors? Can I answer this question with an equation instead of a sentence explaining what I just stated?

Answer 1

The question says "find all vectors". You have found an equation that gives all of them, so you should put "all vectors satisfying $(1,1,1)\cdot\mathbf r=0$" as the answer.