**(i)** State or imply a correct normal vector to either plane, e.g. $\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}$ or $2\mathbf{i} + \mathbf{j} + 3\mathbf{k}$ | B1 |

Carry out correct process for evaluating the scalar product of the two normals | M1 |

Using the correct process for the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result | M1 |

Obtain the final answer $79.75°$ (or $1.39$ radians) | A1 | [4]

**(ii) EITHER:**

Carry out a method for finding a point on the line | M1 |

Obtain such a point, e.g. $(1, 3, 0)$ | A1 |

**EITHER:** State two correct equations for the direction vector $(a, b, c)$ of the line, e.g. $a + 2b - 2c = 0$ and $2a + b + 3c = 0$ | B1 |

Solve for one ratio, e.g. $a : b$ | M1 |

Obtain $a : b : c = 8 : -7 : -3$, or equivalent | A1 |

State a correct final answer, e.g. $\mathbf{r} = \mathbf{i} + 3\mathbf{j} + \lambda(8\mathbf{i} - 7\mathbf{j} - 3\mathbf{k})$ | A1√ |

**OR1:**

Obtain a second point on the line, e.g. $\left(0, \frac{31}{8}, \frac{3}{8}\right)$ | A1 |

Subtract position vectors to find a direction vector | M1 |

Obtain $\mathbf{i} - \frac{7}{8}\mathbf{j} - \frac{3}{8}\mathbf{k}$, or equivalent | A1 |

State a correct final answer, e.g. $\mathbf{r} = \mathbf{i} + 3\mathbf{j} + \lambda\left(-\frac{7}{8}\mathbf{j} - \frac{3}{8}\mathbf{k}\right)$ | A1√ |

**OR2:**

Attempt to calculate the vector product of two normals | M1 |

Obtain two correct components | A1 |

Obtain $8\mathbf{i} - 7\mathbf{j} - 3\mathbf{k}$, or equivalent | A1 |

State a correct final answer, e.g. $\mathbf{r} = \mathbf{i} + 3\mathbf{j} + \lambda(8\mathbf{i} - 7\mathbf{j} - 3\mathbf{k})$ | A1√ |

**OR3:**

Express one variable in terms of a second | M1 |

Obtain a correct simplified expression, e.g. $x = (31 - 8y) / 7$ | A1 |

Express the first variable in terms of a third | M1 |

Obtain a correct simplified expression, e.g. $x = (3 - 3z) / 3$ | A1 |

Form a vector equation of the line | M1 |

State a correct final answer, e.g. $\mathbf{r} = \frac{31}{8}\mathbf{i} + \frac{3}{8}\mathbf{k} + \lambda(8\mathbf{i} - 7\mathbf{j} - 3\mathbf{k})$ | A1√ |

**OR4:**

Express one variable in terms of a second | M1 |

Obtain a correct simplified expression, e.g. $y = (31 - 7x) / 7$ | A1 |

Express the third variable in terms of the second | M1 |

Obtain a correct simplified expression, e.g. $z = (3 - 3x) / 8$ | A1 |

Form a vector equation of the line | M1 |

State a correct final answer, e.g. $\mathbf{r} = \frac{31}{8}\mathbf{i} + \frac{3}{8}\mathbf{k} + \lambda(-8\mathbf{i} + 7\mathbf{j} + 3\mathbf{k})$ | A1√ | [6]

[The f.t. is dependent on all M marks having been earned.]