# Question 7(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mathbf{m} = \mathbf{v}+\frac{1}{2}(\mathbf{w}-\mathbf{v})\Rightarrow$ | M1 | For using vector triangle, or equivalent, for $M$. $\overrightarrow{UM} = \overrightarrow{UV}+\overrightarrow{VM} = (\mathbf{v}-\mathbf{u})+\frac{1}{2}(\mathbf{w}-\mathbf{v})$ |
| $\overrightarrow{UM} = \mathbf{v}+\frac{1}{2}(\mathbf{w}-\mathbf{v})-\mathbf{u} = \frac{1}{2}(\mathbf{v}+\mathbf{w}-2\mathbf{u})$ | A1 | For correct expression AG. SR Allow use of ratio theorem. Minimum $-\mathbf{u}+\frac{1}{2}(\mathbf{v}+\mathbf{w})$ |

## Question 7(ii):

**METHOD 1 (first 3 marks)**

| Answer | Mark | Guidance |
|--------|------|----------|
| $\overrightarrow{UM}$ is $\mathbf{r} = \mathbf{u} + \frac{1}{2}t(\mathbf{v} + \mathbf{w} - 2\mathbf{u})$ | M1* | For equation of $UM$ |
| $t = \frac{2}{3} \Rightarrow \mathbf{u} + \frac{1}{3}(\mathbf{v} + \mathbf{w} - 2\mathbf{u}) = \frac{1}{3}(\mathbf{u} + \mathbf{v} + \mathbf{w})$ | M1*, A1 | For attempt to find suitable value of $t$; for $t = \frac{2}{3}$ and $G$ obtained |

**METHOD 2 (first 3 marks)**

| Answer | Mark | Guidance |
|--------|------|----------|
| $\overrightarrow{UG} = \frac{1}{3}(\mathbf{u} + \mathbf{v} + \mathbf{w}) - \mathbf{u} = \frac{1}{3}(\mathbf{v} + \mathbf{w} - 2\mathbf{u})$ | M1* | For finding directions of $UG$ or $MG$ |
| $\overrightarrow{MG} = \frac{1}{3}(\mathbf{u} + \mathbf{v} + \mathbf{w}) - \frac{1}{2}(\mathbf{v} + \mathbf{w}) = -\frac{1}{6}(\mathbf{v} + \mathbf{w} - 2\mathbf{u})$ | M1* | For comparing with $UM$ |
| $\Rightarrow U, G, M$ collinear | A1 | For showing $G$ lies on $UM$ |

| Answer | Mark | Guidance |
|--------|------|----------|
| By symmetry of $\overrightarrow{OG}$ in $\mathbf{u}, \mathbf{v}, \mathbf{w}$ | B1 | For use of symmetry, or by repeating method for $UM$ twice more |
| $G$ also lies on $VN$, $WP$; $\Rightarrow UM$, $VN$, $WP$ intersect at $G$ | B1dep* | For complete reasoning |

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## Question 7(iii):

| Answer | Mark | Guidance |
|--------|------|----------|
| Line is $\mathbf{r} = \frac{1}{3}(\mathbf{u} + \mathbf{v} + \mathbf{w}) + t(\mathbf{u} - \mathbf{v}) \times (\mathbf{u} - \mathbf{w})$ | B1 | For $r = \frac{1}{3}(\mathbf{u} + \mathbf{v} + \mathbf{w}) + t \times$ "any vector" |
| | B1 | For correct $\mathbf{n}$, using any 2 of $\pm(\mathbf{u}-\mathbf{v}), \pm(\mathbf{v}-\mathbf{w}), \pm(\mathbf{w}-\mathbf{u})$; allow $\overrightarrow{UV} \times \overrightarrow{VW}$ or similar |

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## Question 7(iv):

**METHOD 1**

| Answer | Mark | Guidance |
|--------|------|----------|
| $\mathbf{n} = [1,0,-1] \times [0,1,-1] = k[1,1,1]$ | M1* | For attempt to find $\mathbf{n}$; may see use of $\frac{|\mathbf{p} \cdot \mathbf{n} - d|}{|\mathbf{n}|}$ |
| $UVW$ is $\mathbf{r} \cdot \mathbf{n} = [1,0,0] \cdot [1,1,1] = 1$ | M1dep* | For substituting a point |
| $\Rightarrow d = \frac{1}{\sqrt{3}}$ | A1 | For correct $d$ |

**METHOD 2**

| Answer | Mark | Guidance |
|--------|------|----------|
| $UVW$ is $x + y + z = 1$ | M2 | For attempt to find Cartesian equation |
| $\Rightarrow d = \frac{1}{\sqrt{3}}$ | A1 | For correct $d$ |

**METHOD 3**

| Answer | Mark | Guidance |
|--------|------|----------|
| $\overrightarrow{OG} = \frac{1}{3}(\mathbf{u} + \mathbf{v} + \mathbf{w})$ | M1* | For stating or implying $|\overrightarrow{OG}|$ is $d$ |
| $\Rightarrow OG = \sqrt{\frac{1}{9} + \frac{1}{9} + \frac{1}{9}}$ | M1dep* | For finding magnitude |
| $\Rightarrow d = \frac{1}{\sqrt{3}}$ | A1 | For correct $d$ |

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