Simultaneous Vector Equations

Question

How do I solve the simultaneous vector equations for $r$

$$r \wedge a = b, \qquad r \cdot c = \alpha $$

given that $a\cdot b=0$ and $a$ is not equal to $0$?

I am required to distinguish between the cases $a\cdot c$ is not equal to $0$ and $a\cdot c=0$ and give a geometrical interpretation.

Answer 1

I'll try. Using the property of triple product $c\cdot (r \wedge a) = r \cdot (a \wedge c) = c\cdot b$.

So there are

$$r\cdot b = 0$$ $$r\cdot(a \wedge c) = c\cdot b $$ $$r\cdot c = \alpha $$

If $a \nparallel c$ vectors {$c, a \wedge c, b$} is basis $R^3$.

Applying Gram–Schmidt process we will have orthonormal basis: $e_1 = \frac{c}{|c|}, e_2 = \frac{a \wedge c}{|a \wedge c|}, e_3 = b - (b\cdot e_1)e_1 - (b\cdot e_2)e_2$.

Final $r = \alpha e_1 + \frac{(c\cdot b)}{|a \wedge c|}e_2 + (-\frac{(b\cdot c)}{|c|^2}\alpha -\frac{(b\cdot a \wedge c)}{|a \wedge c|^2}(b\cdot c))e_3$

I just took appropriate basis.