Find the area of the triangle with vertices $A(1,1,2), B(2,3,5)$ and $C(1,5,5)$.

$A$ unit vector perpendicular to the plane of $a = 2i - 6j - 3k$ and $b = 4i + 3j - k$ is

Prove that in a $\Delta ABC$,$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$,where $a, b, c$ represent the magnitudes of the sides opposite to vertices $A, B, C$,respectively.

Let $\hat{u} = u_1 \hat{i} + u_2 \hat{j} + u_3 \hat{k}$ be a unit vector in $\mathbb{R}^3$ and $\hat{v} = \frac{1}{\sqrt{6}}(\hat{i} + \hat{j} + 2 \hat{k})$. Given that there exists a unit vector $\vec{w}$ such that $\hat{u} \times \vec{w} = \hat{v}$,which of the following is(are) correct?

$\text{If } \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2\hat{i} - \hat{j} + 3\hat{k} \text{ and } \vec{c} = \hat{i} - \hat{j} \text{ and if } 6\hat{i} + 2\hat{j} + 3\hat{k} = \lambda_1(\vec{a} \times \vec{b}) + \lambda_2(\vec{b} \times \vec{c}) + \lambda_3(\vec{c} \times \vec{a}), \text{ then } (\lambda_1, \lambda_2, \lambda_3) = $

The magnitude of the projection of the vector $2\hat{i} + 3\hat{j} + \hat{k}$ on the vector perpendicular to the plane containing the vectors $\hat{i} + \hat{j} + \hat{k}$ and $\hat{i} + 2\hat{j} + 3\hat{k}$ is