Let $\vec{a}$ and $\vec{b}$ be two unit vectors. If the vectors $\vec{c} = \vec{a} + 2\vec{b}$ and $\vec{d} = 5\vec{a} - 4\vec{b}$ are perpendicular to each other,then the angle between $\vec{a}$ and $\vec{b}$ is:

If $A=(-2,2,3), B=(3,2,2), C=(4,-3,5)$ and $D=(7,-5,-1)$,then the projection of $\overline{AB}$ on $\overline{CD}$ is

What is the projection vector of the vector $\vec{a} = (1, 1, 1)$ onto the vector $\vec{b} = (2, 2, 1)$?

If $|a| = 3, |b| = 4, |c| = 5$ and $a + b + c = 0,$ then the angle between $a$ and $b$ is

If $a = i + 2j - 3k$ and $b = 3i - j + 2k$,find the angle between the vectors $a + b$ and $a - b$ in degrees.

Let $\bar{a}, \bar{b}$ and $\bar{c}$ be vectors of magnitude $2, 3$ and $4$ respectively. If $\bar{a}$ is perpendicular to $(\bar{b}+\bar{c})$,$\bar{b}$ is perpendicular to $(\bar{c}+\bar{a})$ and $\bar{c}$ is perpendicular to $(\bar{a}+\bar{b})$,then the magnitude of $\bar{a}+\bar{b}+\bar{c}$ is