a) Express general point of the line in component form, e.g. $\left( {2 + \lambda {\text{ }},{\text{ }} - 1 + 2\lambda {\text{ }},{\text{ }} - 4 + 2\lambda } \right)$

Substitute in plane equation and solve for $\lambda $

Obtain position vector $4i + 3j$, or equivalent

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

Using the correct process, evaluate the scalar product of a direction vector for $l$ and a normal for $p$

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

Obtain answer ${26.5^ \circ }$ (or 0.462 radians)

c) EITHER: State $\alpha  + 2b + 2c = 0$ or $3\alpha  - b + 2c = 0$

Obtain two relevant equations and solve for one ratio, e.g. $\alpha :b$

Obtain $\alpha :b:c = 6:4: - 7$, or equivalent

Substitute coordinates of a relevant point in $6x + 4y - 7z = d$ and evaluate $d$

Obtain answer $6x + 4y - 7z = 36$, or equivalent

OR1: Attempt to calculate vector product of relevant vectors,

e.g. $\left( {i + 2j + 2k} \right) \times \left( {3i - j + 2k} \right)$

Obtain two correct components of the product

Obtain correct product, e.g. $6i + 4j - 7k$

Substitute coordinates of a relevant point in $6x + 4y - 7z = d$ and evaluate $d$

Obtain answer $6x + 4y - 7z = 36$, or equivalent

OR2: Attempt to form 2-parameter equation with relevant vectors

State a correct equation, e.g. $r = 2i - j - 4k + \lambda \left( {i + 2j + 2k} \right) + \mu \left( {3i - j + 2k} \right)$

State three equations in  $x$, $y$, $z$, $\lambda $, $\mu $

Eliminate $\lambda $ and $\mu $

Obtain answer $6x + 4y - 7z = 36$, or equivalent