11 questions
Let \( S \) be the reflection of a point \( Q \) with respect to the plane given by \( \vec{r}=-(t+p)\hat{\imath}+t\hat{\jmath}+(1+p)\hat{k} \) where \( t, p \) are real parameters and \( \hat{\imath}, \hat{\jmath}, \hat{k} \) are the unit vectors along the three positive coordinate axes. If the position vectors of \( Q \) and \( S \) are \( 10\hat{\imath}+15\hat{\jmath}+20\hat{k} \) and \( \alpha\hat{\imath}+\beta\hat{\jmath}+\gamma\hat{k} \) respectively, then which of the following is/are TRUE?
Let $P$ be the plane $\sqrt{3}x + 2y + 3z = 16$ and let
$S = \left\{ \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k} : \alpha^2 + \beta^2 + \gamma^2 = 1 \text{ and the distance of } (\alpha, \beta, \gamma) \text{ from the plane } P \text{ is } \frac{7}{2} \right\}.$
Let $\vec{u}, \vec{v}$ and $\vec{w}$ be three distinct vectors in $S$ such that $|\vec{u} - \vec{v}| = |\vec{v} - \vec{w}| = |\vec{w} - \vec{u}|$. Let $V$ be the volume of the parallelepiped determined by vectors $\vec{u}, \vec{v}$ and $\vec{w}$. Then the value of $\frac{80}{\sqrt{3}}V$ is _____.
