If $\vec{a} = -4 \hat{i} + 2 \hat{j} + 4 \hat{k}$ and $\vec{b} = \sqrt{2} \hat{i} - \sqrt{2} \hat{j}$ are two vectors,then the angle between the vectors $2 \vec{a}$ and $\frac{\vec{b}}{2}$ is (in $^{\circ}$)

If $a+b+c=0$ and $|a|=3, |b|=5, |c|=7$,then the angle between $a$ and $b$ is ........ (in $^{\circ}$)

If $a, b$ and $c$ are perpendicular to $b + c, c + a$ and $a + b$ respectively,and if $|a + b| = 6, |b + c| = 8$ and $|c + a| = 10$,then $|a + b + c| = $

If $ai + 6j - k$ and $7i - 3j + 17k$ are perpendicular vectors,then the value of $a$ is

Let $a = \sin^2 x \hat{i} + \cos^2 x \hat{j} + \hat{k}$,where $x \in R$. If the pairs of vectors $(a, \hat{i})$,$(a, \hat{j})$,and $(a, \hat{k})$ are adjacent sides of $3$ distinct parallelograms and $A$ is the sum of the squares of the areas of these parallelograms,then $A$ lies in the interval

If $|\vec{a}|=4, |\vec{b}|=5, |\vec{a}-\vec{b}|=3$ and $\theta$ is the angle between the vectors $\vec{a}$ and $\vec{b}$,then $\cot^2 \theta=$