The angle between two lines whose direction ratios are propotional to $1,1,-2$ and $(\sqrt{3}-1),(-\sqrt{3}-1),-4$ is :

If $\theta$ is angle between $\vec{a}$ & $\vec{b}$, then for vectors $\vec{a}, \vec{b}$ and $\vec{c}$,

Match the LIST-I with LIST-II

Choose the correct answer from the options given below:

Two vectors $\vec{a}$ and $\vec{b}$ are inclined at angle $\theta=60°$ such that $|\vec{a}|=1$, $|\vec{b}|=2$, then $\left|(\vec{a}+3\vec{b})\times(3\vec{a}-\vec{b})\right|^2$ is equal to

For any vector $\vec{a}$, the value of $|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$ is equal to:

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

If $|\vec{a}| = 3|\vec{b}|$, $|\vec{b}| = 2$ and angle between $\vec{a}$ and $\vec{b}$ is $60^\circ$, then $|\vec{a} - \vec{b}|$ is equal to:

If a unit vector $\hat a$ makes an angle $\dfrac{\pi}{3}$ with $\hat i$, $\dfrac{\pi}{4}$ with $\hat j$ and an acute angle $\theta$ with $\hat k$, then the respective value of $\theta$ and $\hat a$ are