Question #40849

Let $M$ be an orthogonal matrix of order 3 with column vectors $\vec{c}_1, \vec{c}_2, \vec{c}_3$. Let $\vec{u} = \vec{c}_1$, $\vec{v} = \vec{c}_1 + \vec{c}_2$, and $\vec{w} = \vec{c}_1 + \vec{c}_2 + \vec{c}_3$. Two planes $P_1$ and $P_2$ have normal vectors $\vec{n}_1$ and $\vec{n}_2$ respectively. The scalar projections of $\vec{n}_1$ onto $\vec{u}, \vec{v}, \vec{w}$ are $1, \sqrt{2}, \sqrt{3}$ respectively. The scalar projections of $\vec{n}_2$ onto $\vec{u}, \vec{v}, \vec{w}$ are $2, 0, -\sqrt{3}$ respectively. If $\theta$ is the acute angle between the planes $P_1$ and $P_2$, then $\cos \theta$ is equal to: