The value of $\lambda$ for which the vectors $\vec{a} = 2\hat{i} + \lambda\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}$ are orthogonal is

In a quadrilateral $ABCD$, $\measuredangle A = \frac{2\pi}{3}$ and $\vec{AC}$ is the bisector of angle $A$. If $15|\vec{AC}| = 5|\vec{AD}| = 3|\vec{AB}|$, then the angle between $\vec{AB}$ and $\vec{BC}$ is

$A$ parallelogram is constructed with $5\vec{a} + 2\vec{b}$ and $\vec{a} - 3\vec{b}$ as its adjacent sides, where $|\vec{a}| = 2\sqrt{2}$ and $|\vec{b}| = 3$. The angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$. Find the lengths of the diagonals of the parallelogram.

Let $\bar{a} = \bar{i} + 2\bar{j} + 3\bar{k}$, $\bar{b} = 2\bar{i} - 3\bar{j} + \bar{k}$, and $\bar{c} = 3\bar{i} + \bar{j} - 2\bar{k}$ be three vectors. If $\bar{r}$ is a vector such that $\bar{r} \cdot \bar{a} = 0$, $\bar{r} \cdot \bar{b} = -2$, and $\bar{r} \cdot \bar{c} = 6$, then find the value of $\bar{r} \cdot (3\bar{i} + \bar{j} + \bar{k})$.

If the angles between the sides of the triangle $ABC$ formed by $A(2,3,5)$,$B(-1,3,2)$ and $C(3,5,-2)$ are $\alpha, \beta$ and $\gamma$,then $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = $

If $4i + 7j + 8k$,$2i + 3j + 4k$ and $2i + 5j + 7k$ are the position vectors of the vertices $A$,$B$ and $C$ respectively of triangle $ABC$. The position vector of the point where the bisector of angle $A$ meets $BC$ is