Given $n\times 3$ matrix $A$ and unit vector $\hat{b}\in R^3$, is there a closed-form solution for $x\in R^3$ that minimizes the Euclidean norm $\Vert Ax \Vert$ subject to $\hat{b}^Tx=0$ and $\Vert x \Vert =1$? 

I tried to solve it using Lagrange multiplier, but to no avail... 

This was my approach:
Define the following Lagrangian function

$L(x,\lambda\_1,\lambda\_2) = x^TA^TAx-2\lambda\_1\hat{b}^Tx-\lambda\_2(x^Tx-1)$

Differentiating wrt each variables and setting it to zero gives:

(1) $\frac{\partial L}{\partial x} = 2A^TAx-2\lambda\_1\hat{b}-2\lambda\_2x=0$

(2) $\frac{\partial L}{\partial \lambda\_1} = \hat{b}^Tx=0$

(3) $\frac{\partial L}{\partial \lambda\_2} = x^Tx-1=0$

Then, left-multiplying both sides of Eq (1) by $\hat{b}^T$ leads to

(4) $\lambda\_1=\hat{b}^TA^TAx$ 

So, Eq (1) becomes

(5) $A^TAx-\hat{b}\hat{b}^TA^TAx-\lambda\_2 x = 0$

Left-multiplying both sides of Eq (5) by x^T leads to

(6) $\lambda\_2=x^TA^TAx$

This is basically where I'm stuck now.