Let $\vec{a} = 2\hat{i} + 3\hat{j} - 4\hat{k}$ and $\vec{b} = 3\hat{i} - 5\hat{j} + 6\hat{k}$. Let $\vec{c}$ be a vector such that $\vec{c} \times \vec{a} = \vec{b} \times \vec{c}$ and $\vec{c}\cdot(2\vec{a}-3\vec{b}) = 238\sqrt{2}$ then $|\vec{c}|^2$ is equal to :

A unit vector perpendicular to the vectors $\hat{i} - \hat{j}$ and $\hat{i} + \hat{j}$ is

If the points $A, B, C$ with position vectors $20\hat{i} + \lambda\hat{j}$, $5\hat{i} - \hat{j}$ and $10\hat{i} - 13\hat{j}$ respectively are collinear, then the value of $\lambda$ is

The vector in the direction of the vector $2\hat{i} - \hat{j} - 2\hat{k}$ that has magnitude 9 units is:

If $\vec{a}$ and $\vec{b}$ are two non-zero vectors such that $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$, then the angle $\theta$ between $\vec{a}$ and $\vec{b}$ is

The projection of the vector $5\hat{i} + \hat{j} - 3\hat{k}$ on the vector $\hat{i} + 2\hat{j} - 3\hat{k}$ is

If $|\vec a|=2, |\vec b|=7$ and $\vec a\times\vec b=3\hat i+2\hat j+6\hat k$, then angle between $\vec a$ and $\vec b$ is: