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 \(0\).