The component of $\vec{i} + \vec{j}$ along $\vec{j} + \vec{k}$ is:

Let $\bar{a}, \bar{b}, \bar{c}, \bar{d}$ be vectors such that $\bar{a} \times \bar{b} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\bar{c} \times \bar{d} = 3\hat{i} + 2\hat{j} + \lambda\hat{k}$. If $\begin{vmatrix} \bar{a} \cdot \bar{c} & \bar{b} \cdot \bar{c} \\ \bar{a} \cdot \bar{d} & \bar{b} \cdot \bar{d} \end{vmatrix} = 0$,then find the value of $\lambda$.

The least positive integral value of $\alpha$,for which the angle between the vectors $\alpha \hat{i}-2 \hat{j}+2 \hat{k}$ and $\alpha \hat{i}+2 \alpha \hat{j}-2 \hat{k}$ is acute,is

If $|\vec{a}|=4, |\vec{b}|=5, |\vec{a}-\vec{b}|=3$ and $\theta$ is the angle between the vectors $\vec{a}$ and $\vec{b}$,then $\cot^2 \theta=$

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

If $|\vec{a}|=3, |\vec{b}|=5$ and $|\vec{c}|=7$ and $\vec{a}+\vec{b}+\vec{c}=\vec{0}$,then the angle between $\vec{a}$ and $\vec{b}$ is