Find $a$ given $a \times b = c$ and $a \cdot d = e$ where $a,b,c,d$ are vectors and $e$ is a scalar

Question

I can't work out how to find $a$? I don't feel as though we have enough information, though apparently we do. Also $b \cdot d\ne0$.

Answer 1

If the solution exists it's unique, as we can't add to $a$ something parallel to $b$ yet orthogonal to $d$. Since $a\perp c$, take the Ansatz $a=(Xb+Yd)\times c$ so$$c=((Xb+Yd)\times c)\times b=X[b^2c-(b\cdot c)b]+Y[(b\cdot d)c-(b\cdot c)d].$$But $b\cdot c=0$, so $c=(b^2X+(b\cdot d)Y)c$, which simplifies to $1=b^2X+(b\cdot d)Y$. Finally,$$e=Xb\times c\cdot d\implies X=\frac{e}{b\times c\cdot d},\,Y=\frac{1-b^2X}{b\cdot d}.$$