Solving a linear equation with a cross product

[SP90]

Homework Statement

Suppose v is a vector satisfying:

$\alpha v + ( a \times v ) = b$

For $\alpha$ a scalar and $a, b$ fixed vectors. Use dot and cross product operations to solve the above for v.

Homework Equations

The unique solutions should be:

$v=\frac{\alpha^{2}b- \alpha (b \times a) + (b \cdot a)a}{\alpha(\alpha^{2}+|a|^{2}})$

I'm having trouble getting there.

The Attempt at a Solution

I get that $( a \times v ) = b - \alpha v$ and then dotting both sides with a and v gives two identities:

$(b - \alpha v) \cdot a = 0$
$(b - \alpha v) \cdot v = 0$

Which rearrange to give

$(b \cdot a) = \alpha (v \cdot a)$
$(b \cdot a) = \alpha (v \cdot v) = \alpha |v|^{2}$

I don't know where to go from here. There's no inverse cross product and I've tried several different combinations of cross and dot products which lead to dead ends, or are just the same equations as those two I've just derived.

Any help would be much appreciated.

 

[tiny-tim]

Hi SP90!

You have three vectors, a b and axb, with the third perpendicular to the other two.

So start by finding the component of v along axb, and the component of v perpendicular to axb.