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.]