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

If $U = [2, -3, 4]$,$X = [0, 2, 3]$,$V = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$ and $Y = \begin{bmatrix} 2 \\ 2 \\ 4 \end{bmatrix}$,then $UV + XY = $?

Let $\bar{a}, \bar{b}, \bar{c}$ be three vectors such that $\bar{a}+\bar{b}+\bar{c}=\bar{0}$,$|\bar{a}|=3$,$|\bar{b}|=4$,and $|\bar{c}|=5$. Then,find the value of $\bar{a} \cdot \bar{b}+\bar{b} \cdot \bar{c}+\bar{c} \cdot \bar{a}$.

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:

If $a, b, c$ are unit vectors satisfying the relation $a+b+\sqrt{3} c=0$, then the angle between $a$ and $b$ is

If $\bar{a}$ and $\bar{b}$ are vectors such that $|\bar{a}+\bar{b}|=\sqrt{29}$ and $\bar{a} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \bar{b}$,then a possible value of $(\bar{a}+\bar{b}) \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})$ is