# Another Simultaneous Vector Equation

Under what condition is there a solution to the simultaneous vector equations: $$\alpha x + \beta y = a, \text{ and } x \wedge y = b,$$ for the vectors $x$ and $y$ in terms of given non-zero scalars $\alpha$ and $\beta$ and non-zero vectors $a$ and $b$?

How can I find the general solution to these equations when this condition is satisfied?

-
 If $\wedge$ is the exterior product in Euclidean space then the last equation says that $x$ and $y$ are in a plane orthogonal to $b$. – Henno Brandsma Feb 12 '12 at 11:48 Is there anything else to help me answer this question you can add? i'm really struggling to understand it. – Euden Feb 12 '12 at 13:24 the other equation just says that $a$ is in the linear span of $x,y$, so this happens if $x$ is a multiple of $y$ and $a$ is also in that line, or, more commonly, when $x,y$ are independent and so span the orthogonal plane to $b$, through $0$, and $a$ is also in that plane. – Henno Brandsma Feb 12 '12 at 13:30 Thankyou for this information. Do you have any idea how to answer this question as i'm baffled. – Euden Feb 12 '12 at 13:41 I assume nobody can answer this? – Euden Feb 13 '12 at 16:10