**(i)** | |
**EITHER** Use scalar product of relevant vectors, or subtract point equations to form two equations in $a,b,c$, e.g. $a - 5b - 3c = 0$ and $a - b - 3c = 0$ | M1* |
State two correct equations in $a,b,c$ | A1 |
Solve simultaneous equations and find one ratio, e.g. $a:c$, or $b = 0$ | M1(dep*) |
Obtain $a:b:c = 3:0:1$, or equivalent | A1 |
Substitute a relevant point in $3x + z = d$ and evaluate $d$ | M1(dep*) |
Obtain equation $3x + z = 13$, or equivalent | A1 |

**OR 1** | |
Attempt calculate vector product of relevant vectors, e.g. $(i - 5j - 3k) \times (i - j - 3k)$ | M2* |
Obtain 2 correct components of the product | A1 |
Obtain correct product, e.g. $12i + 4k$ | A1 |
Substitute a relevant point in $12x + 4z = d$ and evaluate $d$ | M1(dep*) |
Obtain $3x + z = 13$, or equivalent | A1 |

**OR 2** | |
Attempt to form 2-parameter equation for the plane with relevant vectors | M2* |
State a correct equation e.g. $r = 3i - 2j + 4k + \lambda(i - 5j - 3k) + \mu(i - j - 3k)$ | A1 |
State 3 equations in $x, y, z, \lambda$ and $\mu$ | A1 |
Eliminate $\lambda$ and $\mu$ | M1(dep*) |
Obtain equation $3x + z = 13$, or equivalent | A1 | [6]

**(ii)** | |
**EITHER** Find $\vec{CP}$ for a point $P$ on $AB$ with a parameter $t$, e.g. $2i + 3j + 7k + t(-i + j + 3k)$ | B1 $\checkmark$ |
Either: Equate scalar product $\vec{CP} \cdot \vec{AB}$ to zero and form an equation in $t$ | |
Or 1: Equate derivative for $CP^2$ (or $CP$) to zero and form an equation in $t$ | |
Or 2: Use Pythagoras in triangle $CPA$ (or $CPB$) and form an equation in $t$ | |
Solve and obtain correct value of $t$, e.g. $t = -2$ | A1 |
Carry out a complete method for finding the length of $CP$ | M1 |
Obtain answer $3\sqrt{2}$ (4.24), or equivalent | A1 |

**OR 1** | |
State $\vec{AC}$ (or $\vec{BC}$) and $\vec{AB}$ in component form | B1 $\checkmark$ |
Using a relevant scalar product find the cosine of $\angle CAB$ (or $\angle CBA$) | M1 |
Obtain $\cos\angle CAB = -\frac{22}{\sqrt{11}\sqrt{62}}$ or $\cos\angle CBA = \frac{33}{\sqrt{11}\sqrt{117}}$, or equivalent | A1 |
Use trig to find the length of the perpendicular | M1 |
Obtain answer $3\sqrt{2}$ (4.24), or equivalent | A1 |

**OR 2** | |
State $\vec{AC}$ (or $\vec{BC}$) and $\vec{AB}$ in component form | B1 $\checkmark$ |
Using a relevant scalar product find the length of the projection $AC$ (or $BC$) on $AB$ | M1 |
Obtain answer $2\sqrt{11}$ (or), $3\sqrt{11}$ or equivalent | A1 |
Use Pythagoras to find the length of the perpendicular | M1 |
Obtain answer $3\sqrt{2}$ (4.24), or equivalent | A1 |

**OR 3** | |
State $\vec{AC}$ (or $\vec{BC}$) and $\vec{AB}$ in component form | B1 $\checkmark$ |
Calculate their vector product, e.g. $(-2i - 3j - 7k) \times (-i + j + 3k)$ | M1 |
Obtain correct product, e.g. $-2i + 13j - 5k$ | A1 |
Divide modulus of the product by the modulus of $\vec{AB}$ | M1 |
Obtain answer $3\sqrt{2}$ (4.24), or equivalent | A1 |

**OR 4** | |
State two of $\vec{AB}, \vec{BC}$ and $\vec{AC}$ in component form | B1 $\checkmark$ |
Use cosine formula in triangle $ABC$ to find $\cos A$ or $\cos B$ | M1 |
Obtain $\cos A = -\frac{44}{2\sqrt{11}\sqrt{62}}$ or $\cos B = \frac{2\sqrt{11}\sqrt{117}}{2\sqrt{117}}$ | A1 |
Use trig to find the length of the perpendicular | M1 |
Obtain answer $3\sqrt{2}$ (4.24), or equivalent | A1 | [5]

[The f.t. is on $\vec{AB}$]