# Find vector x and scalar λ

1. May 8, 2012

### sharks

1. The problem statement, all variables and given/known data
Find the vector x and the scalar λ which satisfy the equations
$$x \wedge b = b-λc,\; x.c=-2$$where b = (-2, 1, -1) and c = (1, -2, 2)

2. Relevant equations
Vector algebra.

3. The attempt at a solution
First, i worked on x.c=-2
Let vector $x = (x_1, x_2, x_3)$
So, i got the first equation: $x_1-2x_2+2x_3=-2$

Now, working with: $x \wedge b = b-λc$
First, i evaluated the L.H.S. and i found the determinant: $(-x_2-x_3)\hat i - (-x_1+2x_3)\hat j +(x_1+2x_2)\hat k$

Next, i found the R.H.S. and i equated both sides, and added them up to get the second equation: $2x_1+x_2-3x_3=-2-λ$

But how to solve both equations to get x and λ? Maybe i've overlooked something...

I also tried vector algebra on: $x \wedge b = b-λc$ by first doing scalar multiplication by b and then vector multiplication by b.
I got: $$x=\frac{-λ(b\wedge c)}{(b^2-λbc)}$$

Well, i think i got it. I had to look closer at my scalar multiplication, which gives:
$$λ=\frac{b^2}{bc}$$
Then, i just have to use the value of λ to find vector x.
λ=-1 and then it seems that i messed up as in finding x, i got the denominator = 0.
Any suggestions?

EDIT: OK... Instead of scalar and vector multiplication by b, i tried the same steps with c.
I got λ=-1 and x=(-5/6, 5/6, 1/6)
Is it correct?

Last edited: May 8, 2012