If $\vec{a}$ and $\vec{b}$ are two non-zero vectors such that $|\vec{a}\times\vec{b}|^2 + |\vec{a}.\vec{b}|^2 = 100$ and $|\vec{a}| = 5$ then the value of $|\vec{b}|$ is

The area (in sq.units) of a triangle formed by vertices O, A and B where $\vec{OA} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{OB} = -3\hat{i} - 2\hat{j} + \hat{k}$ is

The length of line segment joining the points with position vectors $2\hat{i} - 2\hat{j} + 3\hat{k}$ and $5\hat{i} + 2\hat{j} + 3\hat{k}$ is

The projection of the vector $2\hat{i} - \hat{j} + 3\hat{k}$ on the vector $3\hat{i} + 2\hat{j} + 6\hat{k}$ is

If $|\vec{a}| = 3$ and $|\vec{b}| = 4$, then a value of $\lambda$ for which $\vec{a} + \lambda\vec{b}$ and $\vec{a} - \lambda\vec{b}$ are perpendiculars is.

The equation of the line passing through (-2, 3, 4) and parallel to the vector $2\hat{i} - \hat{j} + \hat{k}$ is:

Let $\vec{a} = \hat{i} + 4\hat{j}$, $\vec{b} = 4\hat{j} + \hat{k}$ and $\vec{c} = \hat{i} - 2\hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{a}$ and $\vec{b}$ such that $\vec{c} \cdot \vec{d} = 16$, then $|\vec{d}|$ is equal to