Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
Are you working in $\mathbb{R}^3$ ? –  Henno Brandsma Feb 12 '12 at 11:12
Yes I believe so. –  Euden Feb 12 '12 at 11:57
I've been looking at books for something similar for hours now and have found nothing on this. How can i answer this question? –  Euden Feb 12 '12 at 14:07
add comment

1 Answer

up vote 0 down vote accepted

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.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.