## Question 9(i):

| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply a correct normal vector to either plane, e.g. $2\mathbf{i} + 3\mathbf{j} - \mathbf{k}$, or $\mathbf{i} - 2\mathbf{j} + \mathbf{k}$ | B1 | |
| Carry out correct process for evaluating the scalar product of two normal vectors | M1 | |
| Using the correct process for the moduli, divide the scalar product of the two normal vectors by the product of their moduli and evaluate the inverse cosine of the result | M1 | |
| Obtain answer $56.9°$ or $0.994$ radians | A1 | |
| **Total** | **4** | |

## Question 9(ii):

| Answer | Mark | Guidance |
|--------|------|----------|
| *EITHER:* Carry out a complete strategy for finding a point on line $l$ | M1 | |
| Obtain such a point, e.g. $(1, 1, 4)$ | A1 | |
| *EITHER:* State a correct equation for a direction vector $a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$ for $l$, e.g. $2a + 3b - c = 0$ | B1 | |
| State a second equation, e.g. $a - 2b + c = 0$, and solve for one ratio, e.g. $a:b$ | M1 | |
| Obtain $a:b:c = 1:-3:-7$, or equivalent | A1 | |
| State a correct answer, e.g. $\mathbf{r} = \mathbf{i} + \mathbf{j} + 4\mathbf{k} + \lambda(\mathbf{i} - 3\mathbf{j} - 7\mathbf{k})$ | A1 | |
| *OR1:* Attempt to calculate the vector product of the two normal vectors | M1 | |
| Obtain two correct components | A1 | |
| Obtain $\mathbf{i} - 3\mathbf{j} - 7\mathbf{k}$, or equivalent | A1 | |
| State a correct answer, e.g. $\mathbf{r} = \mathbf{i} + \mathbf{j} + 4\mathbf{k} + \lambda(\mathbf{i} - 3\mathbf{j} - 7\mathbf{k})$, or equivalent | A1 | |
| *OR2:* Obtain a second point on $l$, e.g. $(0, 4, 11)$ | B1 | |
| Subtract position vectors and obtain a direction vector for $l$ | M1 | |
| Obtain $\mathbf{i} - 3\mathbf{j} - 7\mathbf{k}$, or equivalent | A1 | |
| State a correct answer, e.g. $\mathbf{r} = 4\mathbf{j} + 11\mathbf{k} + \mu(\mathbf{i} - 3\mathbf{j} - 7\mathbf{k})$, or equivalent | A1 | |
| *OR3:* Express one variable in terms of a second | M1 | |
| Obtain a correct simplified expression, e.g. $y = 4 - 3x$ | A1 | |
| Express the third variable in terms of the second | M1 | |
| Obtain a correct simplified expression, e.g. $z = 11 - 7x$ | A1 | |
| Form a vector equation for the line | M1 | |
| State a correct answer, e.g. $\mathbf{r} = 4\mathbf{j} + 11\mathbf{k} + \lambda(\mathbf{i} - 3\mathbf{j} - 7\mathbf{k})$, or equivalent | A1 | |
| *OR4:* Express one variable in terms of a second | M1 | |
| Obtain a correct simplified expression, e.g. $x = \frac{4}{3} - \frac{y}{3}$ | A1 | |
| Express the same variable in terms of the third | M1 | |
| Obtain a correct simplified expression, e.g. $x = \frac{11}{7} - \frac{z}{7}$ | A1 | |
| Form a vector equation for the line | M1 | |
| Obtain a correct answer, e.g. $\mathbf{r} = 4\mathbf{j} + 11\mathbf{k} + \mu(\mathbf{i} - 3\mathbf{j} - 7\mathbf{k})$, or equivalent | A1 | |
| **Total** | **6** | |