If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}|=3, |\vec{b}|=4, |\vec{a}+\vec{b}|=\sqrt{37}, |\vec{a}-\vec{b}|=k$ and the angle between $\vec{a}$ and $\vec{b}$ is $\theta$,then find the value of $\frac{4}{13}(k \sin \theta)^2$.

The constant value $(\lambda + \mu)$ for which the lines $\vec{r} = (2\hat{i} + \hat{j} + \hat{k}) + \lambda(\hat{i} - 2\hat{j})$ and $\vec{r} = (\hat{i} + \hat{j} - 3\hat{k}) + \mu(\hat{j} + 2\hat{k})$ intersect each other is equal to (where $\lambda$ and $\mu$ are parameters).

Let $a, b, c$ be three vectors such that the magnitude of $b$ is twice that of $a$ and the magnitude of $c$ is three times that of $a$. If the angle between each pair of vectors is $\frac{\pi}{3}$ and $|a+b+c|=5$,then $|c|+|a|+|b|=$

If in a $\Delta ABC$,$O$ and $O^{\prime}$ are the incentre and orthocentre respectively,then $\vec{O^{\prime}A} + \vec{O^{\prime}B} + \vec{O^{\prime}C}$ is equal to

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\sqrt{3} \vec{c}=\overrightarrow{0}$,then the angle between $\vec{a}$ and $\vec{b}$ is

The value of $b$ such that the scalar product of the vector $(i + j + k)$ with the unit vector parallel to the sum of the vectors $(2i + 4j - 5k)$ and $(bi + 2j + 3k)$ is $1$,is: