Let P be the plane containing the vectors $(6,6,9)$ and $(7,8,10)$. Find a unit vector that is perpendicular to $(2,-3,4)$ and that lies in the plane P. (Note: all vectors are considered as line segments starting at the origin $(0,0,0)$. In particular the origin lies in the plane P.)

We denote by arcch $:[1,+\infty)\rightarrow\mathbb{R}_+$ the inverse of the hyperbolic cosine. Let $v\in\mathcal{H}$. Show that the set $T_v\mathcal{H}$ of vectors tangent to $\mathcal{H}$ at point $v$ is a vector subspace of $V$ and determine this subspace. Deduce that the restriction of $B$ to $T_v\mathcal{H}$ is an inner product.

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:
(1) $3\sqrt{6}$
(2) $\sqrt{\frac{3}{2}}$
(3) $\sqrt{6}$
(4) $\frac{\sqrt{3}}{2}$