**(i)** State correct equation in any form, e.g. $\mathbf{r} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k} + \lambda(2\mathbf{i} + 2\mathbf{j} - 2\mathbf{k})$ | B1 |
[1 mark total]

**(ii)** EITHER: Equate a relevant scalar product to zero and form an equation in $\lambda$ | M1 |

OR 1: Equate derivative of $OP^2$ (or $OP$) to zero and form an equation in $\lambda$ | M1 |

OR 2: Use Pythagoras in $OAP$ or $OBP$ and form an equation in $\lambda$ | M1 |

State a correct equation in any form | A1 |
Solve and obtain $\lambda = -\frac{1}{2}$ or equivalent | A1 |
Obtain final answer $\overrightarrow{OP} = \frac{2}{3}\mathbf{i} + \frac{5}{3}\mathbf{j} + \frac{1}{3}\mathbf{k}$, or equivalent | A1 |
[4 marks total]

**(iii)** EITHER: State or imply $\overrightarrow{OP}$ is a normal to the required plane | M1 |
State normal vector $2\mathbf{i} + 5\mathbf{j} + 7\mathbf{k}$, or equivalent | A1√ |
Substitute coordinates of a relevant point in $2x + 5y + 7z = d$ and evaluate $d$ | M1 |
Obtain answer $2x + 5y + 7z = 26$, or equivalent | A1 |

OR 1: Find a vector normal to plane $AOB$ and calculate its vector product with a direction vector for the line $AB$ | M1* |
Obtain answer $2\mathbf{i} + 5\mathbf{j} + 7\mathbf{k}$, or equivalent | A1 |
Substitute coordinates of a relevant point in $2x + 5y + 7z = d$ and evaluate $d$ | M1(dep*) |
Obtain answer $2x + 5y + 7z = 26$, or equivalent | A1 |

OR 2: Set up and solve simultaneous equations in $a, b, c$ derived from zero scalar products of $a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$ with (i) a direction vector for line $AB$, (ii) a normal to plane $OAB$ | M1* |
Obtain $a : b : c = 2 : 5 : 7$, or equivalent | A1 |
Substitute coordinates of a relevant point in $2x + 5y + 7z = d$ and evaluate $d$ | M1(dep*) |
Obtain answer $2x + 5y + 7z = 26$, or equivalent | A1 |

OR 3: With $Q(x, y, z)$ on plane, use Pythagoras in $OPQ$ to form an equation in $x, y$ and $z$ | M1* |
Form a correct equation | A1√ |
Reduce to linear form | M1(dep*) |
Obtain answer $2x + 5y + 7z = 26$, or equivalent | A1 |

OR 4: Find a vector normal to plane $AOB$ and form a 2-parameter equation with relevant vectors, e.g. $\mathbf{r} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k} + \lambda(2\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}) + \mu(8\mathbf{i} - 6\mathbf{j} + 2\mathbf{k})$ | M1* |
State three correct equations in $x, y, z, \lambda$ and $\mu$ | A1 |
Eliminate $\lambda$ and $\mu$ | M1(dep*) |
Obtain answer $2x + 5y + 7z = 26$, or equivalent | A1 |
[4 marks total]