**Part (i)**

| $n_1 = [1, 1, 0] \times [1, -5, -2]$ | M1 | For attempting to find vector product of the pair of direction vectors |
|---|---|---|
| $= [-2, 2, -6] = k[1, -1, 3]$ | A1 | For correct $n_1$ |
| Use $(2, 2, 1)$ | M1 | For substituting a point into equation |
| $\Rightarrow r \cdot [-2, 2, -6] = -6 \Rightarrow r \cdot [1, -1, 3] = 3$ | A1, 4 | For correct equation, aef in this form |

**METHOD 2**

| $x = 2 + \lambda + \mu$, $y = 2\lambda - 5\mu$, $z = 1 - 2\mu$ | M1, M1 | For writing as 3 linear equations. For attempting to eliminate $\lambda$ and $\mu$ |
|---|---|---|
| $\Rightarrow x - y + 3z = 3$ | A1 | For correct cartesian equation |
| $\Rightarrow r \cdot [1, -1, 3] = 3$ | A1 | For correct equation, aef in this form |

**Part (ii)**

| For $r = a + tb$ | | |
|---|---|---|

**METHOD 1**

| $b = [1, -1, 3] \times [7, 17, -3]$ | M1 | For attempting to find $n_1 \times n_2$ |
|---|---|---|
| $= k[2, -1, -1]$ | A1∇ | For a correct vector. If from $n_1$ in (i). SR a correct vector may be stated without working. SR for $a = [2, 2, 1]$ stated award M0 |
| e.g. $x, y$ or $z = 0$ in $\begin{cases} x - y + 3z = 3 \\ 7x + 17y - 3z = 21 \end{cases}$ | M1 | For attempting to find a point on the line |
| $\Rightarrow a = \left[0, \frac{3}{2}, \frac{3}{2}\right]$ OR $[3, 0, 0]$ OR $[1, 1, 1]$ | A1∇ | For a correct vector. If from equation in (i). SR a correct vector may be stated without working. SR for $a = [2, 2, 1]$ stated award M0 |
| Line is (e.g.) $r = [1, 1, 1] + t[2, -1, -1]$ | A1∇, 5 | For stating equation of line. If from $a$ and $b$. SR $a = [2, 2, 1]$ stated award M0 |

**METHOD 2**

| Solve $\begin{cases} x - y + 3z = 3 \\ 7x + 17y - 3z = 21 \end{cases}$ by eliminating one variable (e.g. $z$). Use parameter for another variable (e.g. $x$) to find other variables in terms of $t$ | M1, M1 | For attempting to solve equations. For attempting to find parametric solution |
|---|---|---|
| (eg) $y = \frac{3}{2} - \frac{1}{2}t$, $z = \frac{3}{2} - \frac{1}{2}t$ | A1∇, A1∇ | For correct expression for one variable. For correct expression for the other variable. If from equation in (i) for both |
| Line is (eg) $r = \left[0, \frac{3}{2}, \frac{3}{2}\right] + t[2, -1, -1]$ | A1∇ | For stating equation of line. If from parametric solutions |

**METHOD 3**

| eg $x, y$ or $z = 0$ in $\begin{cases} x - y + 3z = 3 \\ 7x + 17y - 3z = 21 \end{cases}$ | M1 | For attempting to find a point on the line |
|---|---|---|
| $\Rightarrow a = \left[0, \frac{3}{2}, \frac{3}{2}\right]$ OR $[3, 0, 0]$ OR $[1, 1, 1]$ | A1∇ | For a correct vector. If from equation in (i). SR a correct vector may be stated without working. SR for $a = [2, 2, 1]$ stated award M0 |
| eg $[3, 0, 0] - [1, 1, 1]$ | M1 | For finding another point on the line and using it with the one already found to find $b$ |
| $b = k[2, -1, -1]$ | A1∇ | For a correct vector. If from equation in (i) |
| Line is (eg) $r = [1, 1, 1] + t[2, -1, -1]$ | A1∇ | For stating equation of line. If from $a$ and $b$ |

**METHOD 4**

| A point on $\Pi_1$ is $[2 + \lambda + \mu, 2\lambda - 5\mu, 1 - 2\mu]$ | M1 | For using parametric form for $\Pi_1$ and substituting into $\Pi_2$ |
|---|---|---|
| On $\Pi_2 \Rightarrow [2 + \lambda + \mu, 2\lambda - 5\mu, 1 - 2\mu] \cdot [7, 17, -3] = 21$ | A1 | For correct unsimplified equation |
| $\Rightarrow \lambda - 3\mu = -1$ | A1 | For correct equation |
| Line is (e.g.) $r = [2, 2, 1] + (3\mu - 1)[1, 1, 0] + \mu[1, -5, -2]$ | M1 | For substituting into $\Pi_1$ for $\lambda$ or $\mu$ |
| $\Rightarrow r = [1, 1, 1]$ or $\left[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}\right] + t[2, -1, -1]$ | A1 | For stating equation of line |

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