$\vec{a} = \hat{i} + \hat{j} + \hat{k}$. A vector $\vec{b}$ has $\vec{a} \times \vec{b} = -\hat{i} + 2\hat{j} - \hat{k}$ and $\vec{a} \cdot \vec{b} = 3$. What is $\vec{b}$?

Explanation

1. 01

   Let $\vec{b} = x\hat{i} + y\hat{j} + z\hat{k}$. Then $\vec{a} \cdot \vec{b} = x + y + z = 3$.

2. 02

   $\vec{a} \times \vec{b} = (z - y)\hat{i} + (x - z)\hat{j} + (y - x)\hat{k}$, so $z - y = -1$, $x - z = 2$ and $y - x = -1$.

3. 03

   So $y = x - 1$ and $z = x - 2$; then $3x - 3 = 3$ gives $x = 2$, and $\vec{b} = 2\hat{i} + \hat{j}$.