**(i)** State or imply a correct normal vector to either plane, e.g. $2\mathbf{i} - \mathbf{j} - 3\mathbf{k}$, or $\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$ | 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 answer $57.7°$ (or 1.01 radians) | A1 | [4]

**(ii)** **EITHER:** Carry out a complete method for finding a point on the line | M1 |

Obtain such a point, e.g. $(2, 0, -1)$ | A1 |

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

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

Obtain $a : b : c = 4 : -7 : 5$, or equivalent | A1 |

State a correct answer, e.g. $\mathbf{r} = 2\mathbf{i} - \mathbf{k} + \lambda(4\mathbf{i} - 7\mathbf{j} + 5\mathbf{k})$ | A1√ |

**OR:** Obtain a second point on the line, e.g. $(0, \frac{7}{2}, -\frac{7}{2})$ | A1 |

Subtract position vectors to obtain a direction vector | M1 |

Obtain $4\mathbf{i} - 7\mathbf{j} + 5\mathbf{k}$, or equivalent | A1 |

State a correct answer, e.g. $\mathbf{r} = 2\mathbf{i} - \mathbf{k} + \lambda(4\mathbf{i} - 7\mathbf{j} + 5\mathbf{k})$ | A1√ |

**OR:** Attempt to calculate the vector product of two normals | M1 |

Obtain two correct components | A1 |

Obtain $4\mathbf{i} - 7\mathbf{j} + 5\mathbf{k}$, or equivalent | A1 |

State a correct answer, e.g. $\mathbf{r} = 2\mathbf{i} - \mathbf{k} + \lambda(4\mathbf{i} - 7\mathbf{j} + 5\mathbf{k})$ | A1√ |

**OR1:** Express one variable in terms of a second | M1 |

Obtain a correct simplified expression, e.g. $x = \frac{14-4y}{7}$ | A1 |

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

Obtain a correct simplified expression, e.g. $x = \frac{14+4z}{5}$ | A1 |

Form a vector equation for the line | M1 |

State a correct answer, e.g. $\mathbf{r} = \frac{2}{7}\mathbf{i} - \frac{2}{7}\mathbf{k} + \lambda(\frac{2}{7}\mathbf{i} + \frac{2}{7}\mathbf{j} + \mathbf{k})$, or equivalent | A1√ |

**OR2:** Express one variable in terms of a second | M1 |

Obtain a correct simplified expression, e.g. $y = \frac{14-7x}{4}$ | A1 |

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

Obtain a correct simplified expression, e.g. $z = \frac{5x-14}{4}$ | A1 |

Form a vector equation for the line | M1 |

State a correct answer, e.g. $\mathbf{r} = \frac{2}{7}\mathbf{i} - \frac{2}{7}\mathbf{k} + \lambda(\frac{2}{7}\mathbf{i} + \frac{2}{7}\mathbf{j} + \mathbf{k})$, or equivalent | A1√ | [6] |

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