Let $V$ be a vector space with $\dim V=n$, and let $x$ be a vector in $V$. Let $\ell=\{tx:t\in\mathbf{R}\}$. Denote $x=(x_1,\ldots,x_n)$. Assuming that the representation of the line $\ell$ is given with respect to the standard basis, I would like to find a change of basis from the standard basis $B$ to $B'$ such that $[x]_{B'}=(1,0,\ldots,0)$, i.e. the line $\ell$ is rotated such that it passes through the origin when $t=0$ and the unit vector $(1,0,\ldots,0)$ when $t=1$. I need help finding a new basis $B'$ that satisfies this condition.
My thoughts so far: The new basis $B'$ must contain the vector $x$; since any vector can be built as a unique linear combination of the basis vectors, we would then have: $$[x]_{B'}=a_1x+\cdots=(1,0,\ldots,0)\implies a_1=1,a_i=0\,\forall i>1$$ Now the trouble is just finding $n-1$ other basis vectors all linearly independent from $x$, and I am not entirely sure how to go about this.
I am fairly new to linear algebra so my experience in this field is limited. Any hints on how to go about finding a suitable new basis $B'$? Thanks in advance.