If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors such that $\vec{a} + 2\vec{b} + 2\vec{c} = \vec{0}$,then $|\vec{a} \times \vec{c}|$ is equal to

Given the following simultaneous equations for vectors $x$ and $y$:
$(i) x + y = a$
$(ii) x \times y = b$
$(iii) x \cdot a = 1$
Then $x = ?, y = ?$

If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$ and $|\overrightarrow{a}|=3, |\overrightarrow{b}|=4$ and $|\overrightarrow{c}|=\sqrt{37}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is:

If $a, b, c$ are coplanar vectors,then

Let $a$ and $b$ be linearly independent vectors such that $|a| = \sqrt{3}$, $|b| = 3$, and $|a - b| = 4$. If $a \times (2i + 2j - k) = (2i + 2j - k) \times b$ and $|(a + b) \cdot (3i + 4j + 2k)| = \sqrt{\lambda}$, then $\lambda = ...$

In a right-angled triangle $ABC$,if the hypotenuse $AB = p$,then $\overline{AB} \cdot \overline{AC} + \overline{BC} \cdot \overline{BA} + \overline{CA} \cdot \overline{CB} = ......$