The line L₁ is represented by ${{\boldsymbol{r}}_1} = \left( {\begin{array}{*{20}{c}} 2\\ 5\\ 3 \end{array}} \right) + s\left( {\begin{array}{*{20}{c}} 1\\ 2\\ 3 \end{array}} \right)$  and the line L₂ by ${{\boldsymbol{r}}_2} = \left( {\begin{array}{*{20}{c}} 3\\ { - 3}\\ 8 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} { - 1}\\ 3\\ { - 4} \end{array}} \right)$ .

The lines L₁ and L₂ intersect at point T. Find the coordinates of T.

evidence of equating vectors     (M1)

e.g. ${L_1} = {L_2}$

for any two correct equations     A1A1

e.g. $2 + s = 3 - t$ , $5 + 2s = - 3 + 3t$ , $3 + 3s = 8 - 4t$

attempting to solve the equations     (M1)

finding one correct parameter $(2 = - 1{\text{, }}t = 2)$     A1

the coordinates of T are $(1{\text{, }}3{\text{, }}0)$     A1     N3

[6 marks]

Those candidates prepared in this topic area answered the question particularly well, often only making some calculation error when solving the resulting system of equations. Curiously, a few candidates found correct values for s and t, but when substituting back into one of the vector equations, neglected to find the z-coordinate of T.