a) State or imply a correct normal vector to either plane, e.g. $i + 2j - 2k$ or $2i + j + 3k$

Carry out correct process for evaluating the scalar product of the two normals

Using the correct process for the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result

Obtain the final answer ${79.7^ \circ }$ (or $1.39$ radians)

b) EITHER: Carry out a method for finding a point on the line

Obtain such a point, e.g. $\left( {1,{\text{ }}3,{\text{ }}0} \right)$

EITHER: State two correct equations for the direction vector $\left( {\alpha ,{\text{ }}b,{\text{ }}c} \right)$ of the line, e.g. $\alpha  + 2b - 2c = 0$ and $2\alpha  + b + 3c = 0$

Solve for one ratio, e.g. $\alpha :b$

Obtain $\alpha :b:c = 8: - 7: - 3$, or equivalent

State a correct final answer, e.g. $r = i + 3j + \lambda \left( {8i - 7j - 3k} \right)$

OR1: Obtain a second point on the line, e.g. $\left( {0,\frac{{31}}{8},\frac{3}{8}} \right)$

Subtract position vectors to find a direction vector

Obtain $i - \frac{7}{8}j - \frac{3}{8}k$, or equivalent

State a correct final answer, e.g. $r = i + 3j + \lambda (i - \frac{7}{8}j - \frac{3}{8}k)$

OR2: Attempt to calculate the vector product of two normals

Obtain two correct components

Obtain $8i - 7j - 3k$, or equivalent

State a correct final answer, e.g. $r = i + 3j + \lambda \left( {8i - 7j - 3k} \right)$

OR3: Express one variable in terms of a second

Obtain a correct simplified expression, e.g. $x = \left( {31 - 8y} \right){\text{ }}/{\text{ }}7$

Express the first variable in terms of a third

Obtain a correct simplified expression, e.g. $x = \left( {3 - 8z} \right)/3$

Form a vector equation of the line

State a correct final answer, e.g. $r = \frac{{31}}{8}j + \frac{3}{8}k + \lambda \left( {8i - 7j - 3k} \right)$

OR4: Express one variable in terms of a second

Obtain a correct simplified expression, e.g. $y = \left( {31 - 7x} \right)/7$

Express the third variable in terms of the second

Obtain a correct simplified expression, e.g. $z = \left( {3 - 3x} \right)/8$

Form a vector equation of the line

State a correct final answer, e.g. $r = \frac{{31}}{8}j + \frac{3}{8}k + \lambda \left( { - 8i + 7j + 3k} \right)$

[The f.t. is dependent on all M marks having been earned.]