If $\vec{a}=\hat{i}+\lambda \hat{j}+2 \hat{k}$ and $\vec{b}=\mu \hat{i}+\hat{j}-\hat{k}$ are orthogonal and $|\vec{a}|=|\vec{b}|$,then $(\lambda, \mu) = $

If $\vec{a} = \hat{i} + 2\hat{j} + 2\hat{k}$,$|\vec{b}| = 5$,and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{6}$,then the area of the triangle formed by these two vectors as two sides is

If $|\vec{a}|=5, |\vec{b}|=13$ and $|\vec{a} \times \vec{b}|=25$. If $\frac{\pi}{2} < \theta < \pi$ where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$,then the value of $\vec{a} \cdot \vec{b}$ is:

If unit vectors $\bar{a}$ and $\bar{b}$ are perpendicular to each other and a unit vector $\bar{c}$ makes an angle $\theta$ with both $\bar{a}$ and $\bar{b}$,and $\bar{c} = \alpha \bar{a} + \beta \bar{b} + r(\bar{a} \times \bar{b})$,then:

Find the angle between the vectors $2\hat{i} + 3\hat{j} + \hat{k}$ and $2\hat{i} - \hat{j} - \hat{k}$.

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{b}=-\hat{i}-8\hat{j}+2\hat{k}$ and $\vec{c}=4\hat{i}+c_2\hat{j}+c_3\hat{k}$ be three vectors such that $\vec{b} \times \vec{a}=\vec{c} \times \vec{a}$. If the angle between the vector $\vec{c}$ and the vector $3\hat{i}+4\hat{j}+\hat{k}$ is $\theta$,then the greatest integer less than or equal to $\tan^2 \theta$ is: