$a, b, c$ are three vectors such that $a + b + c = 0$,$|a| = 1, |b| = 2, |c| = 3$. Then $a \cdot b + b \cdot c + c \cdot a$ is equal to:

Find the angle between the vectors $2 \hat{i}-\hat{j}+\hat{k}$ and $3 \hat{i}+4 \hat{j}-\hat{k}$.

If $a = 2i + j + 2k$ and $b = 5i - 3j + k$,then the projection of $b$ on $a$ is

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 $\bar{a} \cdot \bar{b}+\bar{b} \cdot \bar{c}+\bar{c} \cdot \bar{a}$.

For three unit vectors $\vec{a}, \vec{b}, \vec{c}$ satisfying $|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}=9$ and $|2\vec{a}+k\vec{b}+k\vec{c}|=3$, the positive value of $k$ is:

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| = $