Write down a vector equation for line L .

The line L intersects the x-axis at the point P. Find the x-coordinate of P.

any correct equation in the form ${\boldsymbol{r}} = {\boldsymbol{a}} + t{\boldsymbol{b}}$ (accept any parameter for t)

where a is $\left( {\begin{array}{*{20}{c}} 5\\ { - 4}\\ {10} \end{array}} \right)$ , and b is a scalar multiple of $\left( {\begin{array}{*{20}{c}} 4\\ { - 2}\\ 5 \end{array}} \right)$     A2     N2

e.g. ${\boldsymbol{r}} = \left( {\begin{array}{*{20}{c}}   5 \\   { - 4} \\   {10} \end{array}} \right) + t\left( {\begin{array}{*{20}{c}}   4 \\   { - 2} \\   5 \end{array}} \right){\text{, }}{\boldsymbol{r}} = 5{\boldsymbol{i}} - 4{\boldsymbol{j}} + 10{\boldsymbol{k}} + t( - 8{\boldsymbol{i}} + 4{\boldsymbol{j}} - 10{\boldsymbol{k}})$

Note: Award A1 for the form ${\boldsymbol{a}} + t{\boldsymbol{b}}$ , A1 for $L = {\boldsymbol{a}} + t{\boldsymbol{b}}$ , A0 for ${\boldsymbol{r}} = {\boldsymbol{b}} + t{\boldsymbol{a}}$ .

[2 marks]

recognizing that $y = 0$ or $z = 0$ at x-intercept (seen anywhere)     (R1) 

attempt to set up equation for x-intercept (must suggest $x \ne 0$ )     (M1)

e.g. $L = \left( {\begin{array}{*{20}{c}} x\\ 0\\ 0 \end{array}} \right)$ , $5 + 4t = x$ , $r = \left( {\begin{array}{*{20}{c}} 1\\ 0\\ 0 \end{array}} \right)$

one correct equation in one variable     (A1)

e.g. $- 4 - 2t = 0$ , $10 + 5t = 0$

finding $t = - 2$     A1 

correct working     (A1)

e.g. $x = 5 + ( - 2)(4)$

$x = - 3$ (accept $( - 3{\text{, }}0{\text{, }}0)$)     A1     N3

[6 marks]

In part (a), the majority of candidates correctly recognized the equation that contains the position and direction vectors of a line. However, we saw a large number of candidates who continue to write their equations using "$L =$ ", rather than the mathematically correct "${\boldsymbol{r}} =$ " or "$\left( {\begin{array}{*{20}{c}} x\\ y\\ z \end{array}} \right) =$". r and $\left( {\begin{array}{*{20}{c}}   x \\   y \\   z \end{array}} \right)$ represent vectors, whereas L is simply the name of the line. For part (b), very few candidates recognized that a general point on the x-axis will be given by the vector $\left( {\begin{array}{*{20}{c}} x\\ 0\\ 0 \end{array}} \right)$ . Common errors included candidates setting their equation equal to$\left( {\begin{array}{*{20}{c}} 0\\ 0\\ 0 \end{array}} \right)$  , or $\left( {\begin{array}{*{20}{c}} 1\\ 0\\ 0 \end{array}} \right)$ , or even just the number $0$.

In part (a), the majority of candidates correctly recognized the equation that contains the position and direction vectors of a line. However, we saw a large number of candidates who continue to write their equations using "$L =$ ", rather than the mathematically correct "${\boldsymbol{r}} =$ " or "$\left( {\begin{array}{*{20}{c}} x\\ y\\ z \end{array}} \right) =$". r and $\left( {\begin{array}{*{20}{c}}   x \\   y \\   z \end{array}} \right)$ represent vectors, whereas L is simply the name of the line. For part (b), very few candidates recognized that a general point on the x-axis will be given by the vector $\left( {\begin{array}{*{20}{c}} x\\ 0\\ 0 \end{array}} \right)$ . Common errors included candidates setting their equation equal to$\left( {\begin{array}{*{20}{c}} 0\\ 0\\ 0 \end{array}} \right)$  , or $\left( {\begin{array}{*{20}{c}} 1\\ 0\\ 0 \end{array}} \right)$ , or even just the number $0$.