# Question 8:

## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply a correct normal vector to either plane, e.g. $3\mathbf{i}+\mathbf{j}-\mathbf{k}$ or $\mathbf{i}-\mathbf{j}+2\mathbf{k}$ | B1 | |
| Use correct method to calculate their scalar product | M1 | |
| Show value is zero and planes are perpendicular | A1 | Total: [3] |

## Part (ii):
### EITHER:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Carry out a complete strategy for finding a point on $l$ the line of intersection | M1 | |
| Obtain such a point, e.g. $(0,7,5)$, $(1,0,1)$, $(5/4,-7/4,0)$ | A1 | |
| State two equations for a direction vector $a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$ for $l$, e.g. $3a+b-c=0$ and $a-b+2c=0$ | B1 | |
| Solve for one ratio, e.g. $a:b$ | M1 | |
| Obtain $a:b:c=1:-7:-4$, or equivalent | A1 | |
| State a correct answer, e.g. $\mathbf{r}=7\mathbf{j}+5\mathbf{k}+\lambda(\mathbf{i}-7\mathbf{j}-4\mathbf{k})$ | A1$\checkmark$ | |

### OR1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain a second point on $l$, e.g. $(1,0,1)$ | B1 | |
| Subtract vectors and obtain a direction vector for $l$ | M1 | |
| Obtain $-\mathbf{i}+7\mathbf{j}+4\mathbf{k}$, or equivalent | A1 | |
| State a correct answer, e.g. $\mathbf{r}=\mathbf{i}+\mathbf{k}+\lambda(-\mathbf{i}+7\mathbf{j}+4\mathbf{k})$ | A1$\checkmark$ | |

### OR2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt to find the vector product of the two normal vectors | M1 | |
| Obtain two correct components of the product | A1 | |
| Obtain $\mathbf{i}-7\mathbf{j}-4\mathbf{k}$, or equivalent | A1 | |
| State a correct answer, e.g. $\mathbf{r}=7\mathbf{j}+5\mathbf{k}+\lambda(\mathbf{i}-7\mathbf{j}-4\mathbf{k})$ | A1$\checkmark$ | |

### OR1 (parametric method):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Express one variable in terms of a second variable | M1 | |
| Obtain a correct simplified expression, e.g. $y=7-7x$ | A1 | |
| Express the third variable in terms of the second | M1 | |
| Obtain a correct simplified expression, e.g. $z=5-4x$ | A1 | |
| Form a vector equation for the line | M1 | |
| Obtain a correct equation, e.g. $\mathbf{r}=7\mathbf{j}+5\mathbf{k}+\lambda(\mathbf{i}-7\mathbf{j}-4\mathbf{k})$ | A1$\checkmark$ | |

### OR2 (parametric method):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Express one variable in terms of a second variable | M1 | |
| Obtain a correct simplified expression, e.g. $z=5-4x$ | A1 | |
| Express the same variable in terms of the third | M1 | |
| Obtain a correct simplified expression e.g. $z=(7+4y)/7$ | A1 | |
| Form a vector equation for the line | M1 | |
| Obtain a correct equation, e.g. $\mathbf{r}=\frac{5}{4}\mathbf{i}-\frac{7}{4}\mathbf{j}+\lambda(-\frac{1}{4}\mathbf{i}+\frac{7}{4}\mathbf{j}+\mathbf{k})$ | A1$\checkmark$ | Total: [6] |

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