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.