$a, b, c$ are three vectors such that $|a|=1, |b|=2, |c|=3$ and $b \cdot c=0$. If the projection of $b$ along $a$ is equal to the projection of $c$ along $a$,then $|2a+3b-3c|=$

If $\overrightarrow{A} = 3\hat{i} + \hat{j} + 2\hat{k}$ and $\overrightarrow{B} = 2\hat{i} - 2\hat{j} + 4\hat{k}$ and $\theta$ is the angle between $\overrightarrow{A}$ and $\overrightarrow{B}$,then the value of $\sin \theta$ is

Coordinate planes and the planes $\pi_1, \pi_2, \pi_3$ which are respectively parallel to $YZ, ZX, XY$ planes at distances $a, b, c$,form a rectangular parallelepiped. $d_1$ is a diagonal of the face on the $XY$-plane not passing through the origin,and $d_2$ is a diagonal of plane $\pi_2$ coterminous with $d_1$. If none of the coordinates of the vertices of the parallelepiped are negative and the angle between $d_1$ and $d_2$ is $\theta$,then $\cos \theta=$

If the angle between unit vectors $\vec{a}$ and $\vec{b}$ is $2\theta$,and $|\vec{a} - \vec{b}| < 1$ with $0 \le \theta \le \pi$,then in which interval does $\theta$ lie?

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:

Let $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ be four vectors such that $\vec{a}$ is perpendicular only to $\vec{c}$. If the vector $\vec{b}$ is parallel to $(\vec{c}-\vec{d})$, then $\vec{c}$ is equal to: