How is this equation valid$?$

Question

If $$\vec a×\vec r=\vec b+t\vec a$$ and $$\vec a \cdot\vec r =3$$ where $\vec a =2\hat i+\hat j-\hat k$ and $\vec b=-\hat i-2\hat j+\hat k$ then find $\vec r$.

I have found the $\vec r$.

My question is that $\vec a×\vec r$ is a vector perpendicular to $\vec a$ and $\vec b+t\vec a $ is a vector in the plane of $\vec b$ and $\vec a $ which is also parallel to $\vec a$. Then how can both of these be equal$?$

Answer 1

The vector in question is not parallel to $\vec a$ if $\vec b\ne 0$; you can check this on your numerical answer. You might be thinking of the line $\{\vec b+t\vec a\colon t\in\Bbb R\}$, which is indeed a line parallel to $\vec a$. But the individual points on this line, when considered as vectors, are not parallel to $\vec a$.