Find a basis for this set of vectors

Question

Find a basis in $\mathbb{R^3}$ for the set of vectors on the line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$

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I don't know from where should I start. I am supposing that the vector space is $\mathbb{R^3}$ and that the field is $\mathbb{R}$.

I think I have to find a set (say $A$) whose elements are linearly independent and $\operatorname{span}A=B$ where $B$ is the set of all point on the line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$

I don't know if I am going in the right direction and even if I am going in the right direction I can't think of the method to find a basis. Till now, I only solved problems where we have to check whether a set is the basis or not.

Any help is greatly appreciated.

Answer 1

Your set is the set $S$ of all vectors of the form $(2a,3a,4a)$, for some $a\in\Bbb R$. Therefore, $\{(2,3,4)\}$ (for instance) is a basis of $S$.