Let $S$ be the reflection of a point $Q$ with respect to the plane given by $\vec{r}=-(t+p)\hat{i}+t\hat{j}+(1+p)\hat{k}$ where $t$, $p$ are real parameters and $\hat{i}$, $\hat{j}$, $\hat{k}$ are the unit vectors along the three positive coordinate axes. If the position vectors of $Q$ and $S$ are $10\hat{i}+15\hat{j}+20\hat{k}$ and $\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}$ respectively, then which of the following is/are TRUE ?

$\vec r = -(t+p)\hat{i} + t\hat{j} + (1+p)\hat{k}$ $Q = 10\hat{i} + 15\hat{j} + 20\hat{k},\quad S = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$ $\text{Find: which of (1)--(4) is/are TRUE}$ $x = -(t+p),\quad y = t,\quad z = 1+p$ $x + y + z = -(t+p) + t + (1+p) = 1$ $1.\ \vec n = (1,1,1),\ \vec n \cdot Q = ?$ $2.\ F = \text{foot of } \perp \text{ from } Q$ $3.\ S = 2F - Q$ $n = (1,1,1),\quad d = 1$ $n \cdot Q = 10 + 15 + 20 = 45$ $F = Q - \dfrac{n \cdot Q - d}{|n|^2}\,n$ $= Q - \dfrac{44}{3}(1,1,1)$ $F = \left(10 - \dfrac{44}{3},\ 15 - \dfrac{44}{3},\ 20 - \dfrac{44}{3}\right)$ $F = \left(-\dfrac{14}{3},\ \dfrac13,\ \dfrac{16}{3}\right)$ $S = 2F - Q$ $S = \left(-\dfrac{58}{3},\ -\dfrac{43}{3},\ -\dfrac{28}{3}\right)$ $\alpha = -\dfrac{58}{3},\quad \beta = -\dfrac{43}{3},\quad \gamma = -\dfrac{28}{3}$ $3(\alpha+\beta) = 3\left(-\dfrac{58}{3}-\dfrac{43}{3}\right) = 3\left(-\dfrac{101}{3}\right) = -101$ $\Rightarrow (1) \text{ TRUE}$ $3(\beta+\gamma) = 3\left(-\dfrac{43}{3}-\dfrac{28}{3}\right) = 3\left(-\dfrac{71}{3}\right) = -71$ $\Rightarrow (2) \text{ TRUE}$ $3(\gamma+\alpha) = 3\left(-\dfrac{28}{3}-\dfrac{58}{3}\right) = 3\left(-\dfrac{86}{3}\right) = -86$ $\Rightarrow (3) \text{ TRUE}$ $3(\alpha+\beta+\gamma) = 3\left(-\dfrac{129}{3}\right) = -129 \neq -121$ $\Rightarrow (4) \text{ FALSE}$ $\boxed{(1),\ (2),\ (3) \text{ are TRUE}}$