The angle between the vectors $3\,i + j + 2\,k$ and $2\,i - 2\,j + 4\,k$ is

If three vectors $a, b, c$ satisfy $a + b + c = 0$ and $|a| = 3, |b| = 5, |c| = 7,$ then the angle between $a$ and $b$ is .............. $^o$

If $a = 4i + 6j$ and $b = 3j + 4k$,then the component of $a$ along $b$ is

Let $P$ be a real number and $|P| \geq 2$. If $A, B, C$ are variable angles such that $(\sqrt{P^2-4}) \tan A + P \tan B + (\sqrt{P^2+4}) \tan C = 6P$,then the minimum value of $\tan^2 A + \tan^2 B + \tan^2 C$ is:

The value of $\hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{k} \cdot(\hat{i} \times \hat{j})$ is . . . . . . .

If the moduli of the vectors $a, b, c$ are $3, 4, 5$ respectively and $a$ and $b + c$,$b$ and $c + a$,$c$ and $a + b$ are mutually perpendicular,then the modulus of $a + b + c$ is