**(i)** **EITHER:** Obtain a vector parallel to the plane, e.g. $\overrightarrow{AB} = -2\mathbf{i} + 4\mathbf{j} - \mathbf{k}$ | B1 | 

Use scalar product to obtain an equation in $a, b, c$, e.g. $-2a+4b-c=0$, $3a-3b+3c=0$, or $a+b+2c=0$ | M1 | 

Obtain two correct equations in $a, b, c$ | A1 | 

Solve to obtain ratio $a : b : c$ | M1 | 

Obtain $a : b : c = 3:1:-2$, or equivalent | A1 | 

Obtain equation $3x + y - 2z = 1$, or equivalent | A1 | 

**OR1:** Substitute for two points, e.g. $A$ and $B$, and obtain $2a-b+2c=d$ and $3b+c=d$ | B1 | 

Substitute for another point, e.g. $C$, to obtain a third equation and eliminate one unknown entirely from the three equations | M1 | 

Obtain two correct equations in three unknowns, e.g. in $a, b, c$ | A1 | 

Solve to obtain their ratio, e.g. $a : b : c$ | M1 | 

Obtain $a : b : c = 3:1:-2$, $a:c:d = 3:-2:1$ or $b:c:d = 1:-2:1$, or equivalent | A1 | 

Obtain equation $3x + y - 2z = 1$, or equivalent | A1 | 

**OR2:** Obtain a vector parallel to the plane, e.g. $\overrightarrow{AC} = \mathbf{i} + \mathbf{j} + 2\mathbf{k}$ | B1 | 

Obtain a second such vector and calculate their vector product e.g. $(-2\mathbf{i}+4\mathbf{j}-\mathbf{k})\times(3\mathbf{i}-3\mathbf{j}+3\mathbf{k})$ | M1 | 

Obtain two correct components of the product | A1 | 

Obtain correct answer, e.g. $9\mathbf{i} + 3\mathbf{j} - 6\mathbf{k}$ | A1 | 

Substitute in $9x + 3y - 6z = d$ to find $d$ | M1 | 

Obtain equation $3x + y - 2z = 1$, or equivalent | A1 | 

**OR3:** Obtain a vector parallel to the plane, e.g. $\overrightarrow{AC} = \mathbf{i} + \mathbf{j} + 2\mathbf{k}$ | B1 | 

Obtain a second such vector and form correctly a 2-parameter equation for the plane | M1 | 

Obtain a correct equation, e.g. $\mathbf{r} = 3\mathbf{i} + 4\mathbf{k} + \lambda(-2\mathbf{i}+4\mathbf{j}-\mathbf{k}) + \mu(\mathbf{i}+\mathbf{j}+2\mathbf{k})$ | A1 | 

State three correct equations in $x, y, z, \lambda, \mu$ | A1 | 

Eliminate $\lambda$ and $\mu$ | M1 | 

Obtain equation $3x + y - 2z = 1$, or equivalent | A1 | [6]

**(ii)** Obtain answer $\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$, or equivalent | B1 | [1]

**(iii)** **EITHER:** Use $\frac{\overrightarrow{OA}.\overrightarrow{OD}}{|\overrightarrow{OD}|}$ to find projection $ON$ of $OA$ onto $OD$ | M1 | 

Obtain $ON = \frac{4}{3}$ | A1 | 

Use Pythagoras in triangle $OAN$ to find $AN$ | M1 | 

Obtain the given answer | A1 | 

**OR1:** Calculate the vector product of $\overrightarrow{OA}$ and $\overrightarrow{OD}$ | M1 | 

Obtain answer $6\mathbf{i} + 2\mathbf{j} - 5\mathbf{k}$ | A1 | 

Divide the modulus of the vector product by the modulus of $\overrightarrow{OD}$ | M1 | 

Obtain the given answer | A1 | 

**OR2:** Taking general point $P$ of $OD$ to have position vector $\lambda(2\mathbf{i}+2\mathbf{j}+2\mathbf{k})$, form an equation in $\lambda$ by either equating the scalar product of $\overrightarrow{AP}$ and $\overrightarrow{OP}$ to zero, or using Pythagoras in triangle $OPA$, or setting the derivative of $|\overrightarrow{AP}|$ to zero | M1 | 

Solve and obtain $\lambda = \frac{4}{9}$ | A1 | 

Carry out method to calculate $AP$ when $\lambda = \frac{4}{9}$ | M1 | 

Obtain the given answer | A1 | 

**OR3:** Use a relevant scalar product to find the cosine of $AOD$ or $ADO$ | M1 | 

Obtain $\cos AOD = \frac{4}{9}$ or $\cos ADO = \frac{5}{3\sqrt{10}}$, or equivalent | A1 | 

Use trig to find the length of the perpendicular | M1 | 

Obtain the given answer | A1 | 

**OR4:** Use the cosine formula in triangle $AOD$ to find $\cos AOD$ or $\cos ADO$ | M1 | 

Obtain $\cos AOD = \frac{8}{18}$ or $\cos ADO = \frac{10}{6\sqrt{10}}$, or equivalent | A1 | 

Use trig to find the length of the perpendicular | M1 | 

Obtain the given answer | A1 | [4]