Aussie Maths & Science Teachers: Save your time with SmarterEd
Line 1 is given by the equations \(x=-1+2 s, \ y=1-2 s\) and \(z=1+2 s\), where \(s\) is a parameter.
Line 2 is given by the equations \(x=1+2 t, \ y=-1-t\) and \(z=4+3 t\), where \(t\) is a parameter.
Show that Line 1 and Line 2 are skew. (3 marks)
--- 10 WORK AREA LINES (style=lined) ---
\(\text{Skew lines do not intersect and are not parallel.}\)
\(\text{Solving for \(s\) and \(t\) in \(x\) and \(y\):}\)
| \(\text{From} \ x: \quad\) | \(-1+2 s\) | \(=1+2 t\ \ldots\ (1)\) |
| \(\text{From} \ y: \quad\) | \(1-2 s\) | \(=-1-t\ \ldots\ (2)\) |
| \((1)+(2)\) | \(0\) | \(=t\) |
\(\text{Substitute}\ \ t=0\ \ \text{into (1):}\)
\(-1+2 s=1\ \ \Rightarrow\ \ s=1\)
\(\text{Substitute}\ \ s=1\ \ \text{and}\ \ t=0\ \ \text{into}\ z:\)
\(z=1+2=3\ \text{(line 1)}, \ z=4\ \text{(line 2)}\)
\(\Rightarrow\ \text{Lines 1 and 2 do not intersect.}\)
\(\text{Direction vectors:}\)
\(\text{Line 1 = }\left( \begin{array}{r}2 \\ -2 \\ 2\end{array}\right),\ \ \text{Line 2 = } \left(\begin{array}{r}2 \\ -1 \\ 3\end{array}\right).\)
\(\left(\begin{array}{r}2 \\ -2 \\ 2\end{array}\right) \neq k\left(\begin{array}{r}2 \\ -1 \\ 3\end{array}\right) \ \text{for any}\ \ k \in R\)
\(\Rightarrow\ \text{Lines 1 and 2 are not parallel.}\)
\(\therefore\ \text{Lines 1 and 2 are skew as they do not intersect and are not parallel.}\)