Two lines have equations $$\(\mathbf{r}=\left(\begin{array}{l}1 \\ 3 \\ 2\end{array}\right)+s\left(\begin{array}{r}2 \\ -1 \\ 3\end{array}\right)\)$$ and $$\(\mathbf{r}=\left(\begin{array}{l}2 \\ 1 \\ 4\end{array}\right)+t\left(\begin{array}{r}1 \\ -1 \\ 4\end{array}\right)\)$$. Show that the lines are skew. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_m21_qp_32 Year：2021 Question No：7(a)
Answer:
Express general point of a line in component form, e.g.
\((1+2 s, 3-s, 2+3 s) \text { or }(2+t, 1-t, 4+4 t)\)
Equate at least two pairs of components and solve for \(s\) or for \(t\)
Obtain correct answer for \(s\) or for \(t\) (possible answers are \(-1,6, \frac{2}{5}\) for \(s\) and \(-3,4,-\frac{1}{5}\) for \(\left.t\right)\)
Verify that all three component equations are not satisfied
Show that the lines are not parallel and are thus skew
\((1+2 s, 3-s, 2+3 s) \text { or }(2+t, 1-t, 4+4 t)\)
Equate at least two pairs of components and solve for \(s\) or for \(t\)
Obtain correct answer for \(s\) or for \(t\) (possible answers are \(-1,6, \frac{2}{5}\) for \(s\) and \(-3,4,-\frac{1}{5}\) for \(\left.t\right)\)
Verify that all three component equations are not satisfied
Show that the lines are not parallel and are thus skew
Knowledge points:
3.7.4 understand the significance of all the symbols used when the equation of a straight line is expressed in the form r = a +tb, and find the equation of a line, given sufficient information
3.7.5 determine whether two lines are parallel, intersect or are skew, and find the point of intersection of two lines when it exists (Calculation of the shortest distance between two skew lines is not required. Finding the equation of the common perpendicular to two skew lines is also not required.)
Solution:
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