If $|\vec{a}| = 5$, $|\vec{b}| = 6$ and $\vec{a}.\vec{b} = 18$, then value of $|\vec{a} \times \vec{b}|$ is

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 :

If $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}$, $\vec{a} \times \vec{b} = \vec{a} \times \vec{c}$ and $\vec{a} \neq {0}$, then the vector $\vec{b}$ in equal to.

If $\vec a\times\vec b=\vec a\times\vec c$ and $\vec a\neq\vec 0$, then which of the following is/are TRUE?

A. $\vec b=\vec c$

B. $\vec b+\vec c=\vec 0$

C. $\vec b=\vec c+\lambda\vec a$ for some scalar λ

D. $\vec b+\vec c=\lambda\vec a$ for some scalar λ

Choose the correct answer from the options given below:

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

If $\vec a=\hat i+\hat j+2\hat k$ and $\vec b=2\hat i+\hat j+2\hat k$, then the unit vector in the direction of $2\vec a-\vec b$ is equal to :

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