# Question 8:

## Part (a):

| Working/Answer | Mark | Guidance |
|---|---|---|
| $\mathbf{r} = \begin{pmatrix}-4\\-5\\3\end{pmatrix} + t\begin{pmatrix}3\\4\\-1\end{pmatrix}$ | M1 | Forms the parametric form of the line |
| $3(3t-4) + 4(4t-5) - (3-t) = 17 \Rightarrow t = 2$ | dM1 | Substitutes parametric form into plane equation and solves for $t$; depends on first mark |
| $\begin{pmatrix}-4\\-5\\3\end{pmatrix} + \text{"2"}\begin{pmatrix}3\\4\\-1\end{pmatrix}$ | dM1 | Uses their value of $t$ correctly to find $Q$; depends on previous mark |
| $(2, 3, 1)$ | A1 | Correct coordinates; accept column vector but not with **i**, **j**, **k** |

**Way 2:**

| Working/Answer | Mark | Guidance |
|---|---|---|
| $\frac{x+4}{3} = \frac{y+5}{4} = \frac{z-3}{-1}$, rearranges to get 2 of $x,y,z$ as functions of the third | M1 | Forms Cartesian equation of line |
| Substitutes into plane equation and solves for one coordinate | dM1 | |
| Obtains the other 2 coordinates | dM1 | |
| $(2, 3, 1)$ | A1 | Correct coordinates |

## Part (b):

| Working/Answer | Mark | Guidance |
|---|---|---|
| $\mathbf{PQ} = \begin{pmatrix}6\\8\\-2\end{pmatrix}$, $\mathbf{PR} = \begin{pmatrix}3\\11\\1\end{pmatrix}$, $\mathbf{RQ} = \begin{pmatrix}3\\-6\\-3\end{pmatrix}$ (attempts 2 vectors in plane $PQR$) | M1 | Must use given coordinates of $P$, $R$ and their coordinates of $Q$ |
| $\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\6 & 8 & -2\\3 & 11 & 1\end{vmatrix} = \begin{pmatrix}30\\-12\\42\end{pmatrix}$ | dM1 | Attempt vector product between 2 vectors in $PQR$; depends on first mark |
| $\begin{pmatrix}5\\-2\\7\end{pmatrix} \cdot \begin{pmatrix}2\\3\\1\end{pmatrix} = 11$ | dM1 | Uses any of $P$, $Q$ or $R$ to find constant; depends on previous mark |
| $5x - 2y + 7z = 11$ | A1 | Any correct Cartesian equation |

**Way 2:**

| Working/Answer | Mark | Guidance |
|---|---|---|
| $-4a - 5b - 3c = 1$; $2a + 3b + c = 1$; $-a + 6b + 4c = 1$ | M1 | Uses Cartesian form $ax+by+cz=1$ and substitutes coordinates of each of the 3 points |
| Solves to get a value for any of $a$, $b$ or $c$ | dM1 | |
| Obtains values for the other 2 | dM1 | |
| $\frac{5}{11}x - \frac{2}{11}y + \frac{7}{11}z = 1$ | A1 | Any correct Cartesian equation |

## Part (c):

| Working/Answer | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix}-4\\-5\\3\end{pmatrix} + 2\times\text{"2"}\begin{pmatrix}3\\4\\-1\end{pmatrix} = \begin{pmatrix}8\\11\\-1\end{pmatrix}$ | M1 | Correct strategy for another point on $l_3$ |
| $\begin{pmatrix}8\\11\\-1\end{pmatrix} - \begin{pmatrix}-1\\6\\4\end{pmatrix} = \begin{pmatrix}9\\5\\-5\end{pmatrix}$ | dM1 | Attempts direction of $l_3$; depends on first mark |
| $\mathbf{r} = \begin{pmatrix}-1\\6\\4\end{pmatrix} + \lambda\begin{pmatrix}9\\5\\-5\end{pmatrix}$ | ddM1, A1 | ddM1: Forms equation of $l_3$ using $R$ and their direction; depends on both previous marks. A1: Any correct equation in vector form |

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