9758/2022/P1/Q11

11) A gas company has plans to install a pipeline from a gas field to a storage facility. One part of the route for the pipeline has to pass under a river. This part of the pipeline is in a straight line between two points, $P$ and $Q$.

Points are defined relative to an origin $(0,0,0)$ at the gas field. The $x-, y$ – and $z$-axes are in the directions east, north and vertically upwards respectively, with units in metres. $P$ has coordinates $(1136,92, p)$ and $Q$ has coordinates $(200,20,-15)$.
(a) The length of the pipeline $P Q$ is $939 \mathrm{~m}$. Given that the level of $P$ is below that of $Q$, find the value of $p$.

[3]

A thin layer of rock lies below the ground. This layer is modelled as a plane. Three points in this plane are $(400,600,-20),(500,200,-70)$ and $(600,-340,-50)$.
(b) Find the cartesian equation of this plane.

[4]

(d) Find the angle that the pipeline between the points $P$ and $Q$ makes with the horizontal.

[2]

(a)
$$
\begin{array}{l}
\overrightarrow{P Q}=\left(\begin{array}{c}
200 \\
20 \\
-15
\end{array}\right)-\left(\begin{array}{c}
1136 \\
92 \\
p
\end{array}\right)=\left(\begin{array}{c}
-936 \\
-72 \\
-15-p
\end{array}\right) \\
|\overrightarrow{P Q}|=\sqrt{936^2+72^2+(-15-p)^2} \\
939=\sqrt{936^2+72^2+(-15-p)^2}
\end{array}
$$
Using GC, $p=-36$ or $p=6$.
Since $\mathrm{P}$ is below that of $\mathrm{Q} \cdot p=-36$.

(b)
$$
\begin{array}{l}
\text { Normal vector of plane }=\left(\begin{array}{c}
100 \\
-400 \\
-50
\end{array}\right) \times\left(\begin{array}{c}
100 \\
-540 \\
20
\end{array}\right)=\left(\begin{array}{c}
-35000 \\
-7000 \\
-14000
\end{array}\right)=-7000\left(\begin{array}{l}
5 \\
1 \\
2
\end{array}\right) \\
\mathbf{r} \cdot\left(\begin{array}{l}
5 \\
1 \\
2
\end{array}\right)=\left(\begin{array}{c}
400 \\
600 \\
-20
\end{array}\right) \cdot\left(\begin{array}{l}
5 \\
1 \\
2
\end{array}\right) \\
5 x+y+2 z=2560
\end{array}
$$

(c)
$$
\overrightarrow{P Q}=\left(\begin{array}{c}
-936 \\
-72 \\
21
\end{array}\right)=3\left(\begin{array}{c}
312 \\
24 \\
-7
\end{array}\right)
$$
line $P Q: r=\left(\begin{array}{c}200 \\ 20 \\ -15\end{array}\right)+\lambda\left(\begin{array}{c}312 \\ 24 \\ -7\end{array}\right)$ where $\lambda \in \mathbb{R}$
$$
\begin{aligned}
\left(\begin{array}{c}
200+312 \lambda \\
20+24 \lambda \\
-15-7 \lambda
\end{array}\right) \cdot\left(\begin{array}{l}
5 \\
1 \\
2
\end{array}\right) &=2560 \\
990+1570 \lambda &=2560 \\
\lambda &=1
\end{aligned}
$$
$\therefore p+$ of intersection: $\left(\begin{array}{c}512 \\ 44 \\ -22\end{array}\right)$
$\therefore$ coordinate : $(512,44,-22)$

(d)
let \& be $\theta$
$$
\begin{aligned}
\sin \theta &=\frac{\left|\left(\begin{array}{c}
312 \\
24 \\
-7
\end{array}\right) \cdot\left(\begin{array}{l}
0 \\
0 \\
1
\end{array}\right)\right|}{\left|\left(\begin{array}{c}
312 \\
24 \\
-7
\end{array}\right) \right|} \\
\sin \theta &=\frac{7}{313} \\
\theta &=1.28^{\circ} \\
& \approx 1.3^{\circ}
\end{aligned}
$$