Let $\bar{u}, \bar{v}, \bar{w}$ be three vectors such that $|\bar{u}| = 1, |\bar{v}| = 2, |\bar{w}| = 3$. If the projection of $\bar{v}$ along $\bar{u}$ is equal to the projection of $\bar{w}$ along $\bar{u}$ and $\bar{v}, \bar{w}$ are perpendicular to each other, then $|\bar{u} - \bar{v} + \bar{w}| = $...

Let $\bar{a}, \bar{b}, \bar{c}$ be three vectors of equal magnitude such that the angle between $\bar{a}$ and $\bar{b}$ is $\alpha$, $\bar{b}$ and $\bar{c}$ is $\beta$, $\bar{c}$ and $\bar{a}$ is $\gamma$.
Then the minimum value of $\cos\alpha + \cos\beta + \cos\gamma$ is ...

A vector which is orthogonal to the vector $\bar{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and coplanar with the vectors $\bar{b} = 3\hat{i} + 2\hat{j}$ and $\bar{c} = 2\hat{i} + \hat{j} + 3\hat{k}$ is

Let $\vec{a}$ and $\vec{b}$ be linearly independent vectors such that
$|\vec{a}| = \sqrt{3}, |\vec{b}| = 3$ and $|\vec{a} - \vec{b}| = 4$.
If $\vec{a} \times (2\hat{i} + 2\hat{j} - \hat{k}) = (2\hat{i} + 2\hat{j} - \hat{k}) \times \vec{b}$ and $|(\vec{a} + \vec{b}) \cdot (3\hat{i} + 4\hat{j} + 2\hat{k})| = \sqrt{\lambda}$, then $\lambda = \ldots$