The value of $\lambda$ for which the vectors $2\lambda \hat{i} + \hat{j} - \hat{k}$ and $2\hat{j} + \hat{k}$ are perpendicular is:

Let $\vec{u} = 2 \hat{i} + \hat{j}$ and $\vec{v} = 3 \hat{i} - 5 \hat{j}$. Consider three points $P, Q,$ and $R$ having the position vectors $\left(\frac{5}{2}\right) \hat{i} - 2 \hat{j}, \left(\frac{7}{3}\right) \hat{i} - \hat{j},$ and $\left(\frac{9}{4}\right) \hat{i}$ respectively. Among these,the points on the line passing through $\vec{u}$ and $\vec{v}$ are

If vectors $\vec{a}$ and $\vec{b}$ have the same magnitude, the angle between them is $60^\circ$, and their scalar product is $1/2$, then $|\vec{a}|$ is:

If $\overline{a}$ and $\overline{b}$ are two unit vectors such that $5 \overline{a} + 4 \overline{b}$ and $\overline{a} - 2 \overline{b}$ are perpendicular to each other,then the angle between $\overline{a}$ and $\overline{b}$ is

The shortest distance between the lines $r = 3i + 5j + 7k + \lambda(i + 2j + k)$ and $r = -i - j - k + \mu(7i - 6j + k)$ is

The point of intersection of $r \times a = b \times a$ and $r \times b = a \times b$,where $a = i + j$ and $b = 2i - k$ is