Find a vector equation of the line L passing through the points A and B.

Find the coordinates of the point of intersection of the line L with the plane Π.

$\overrightarrow {{\text{AB}}}  = \left( {\begin{array}{*{20}{c}}  6 \\   { - 8} \\   {17} \end{array}} \right)$     (A1)

r = $\left( {\begin{array}{*{20}{c}}  0 \\   3 \\   { - 6} \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}  6 \\   { - 8} \\   {17} \end{array}} \right)$ or r = $\left( {\begin{array}{*{20}{c}}  6 \\   { - 5} \\   {11} \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}  6 \\   { - 8} \\   {17} \end{array}} \right)$     M1A1

Note:     Award M1A0 if r = is not seen (or equivalent).

[3 marks]

substitute line L in $\Pi :4(6\lambda ) - 3(3 - 8\lambda ) + 2( - 6 + 17\lambda ) = 20$     M1

$82\lambda  = 41$

$\lambda  = \frac{1}{2}$     (A1)

r = $\left( {\begin{array}{*{20}{c}}  0 \\   3 \\   { - 6} \end{array}} \right) + \frac{1}{2}\left( {\begin{array}{*{20}{c}}  6 \\   { - 8} \\   {17} \end{array}} \right) = \left( {\begin{array}{*{20}{c}}  3 \\   { - 1} \\   {\frac{5}{2}} \end{array}} \right)$

so coordinate is $\left( {3,{\text{ }} - 1,{\text{ }}\frac{5}{2}} \right)$     A1

Note:     Accept coordinate expressed as position vector $\left( {\begin{array}{*{20}{c}}  3 \\   { - 1} \\   {\frac{5}{2}} \end{array}} \right)$.

[3 marks]