Question

Show that the four points P, Q, R, S with position vectors \[\vec{p}\], \[\vec{q}\], \[\vec{r}\], \[\vec{s}\] respectively such that 5 \[\vec{p}\] − 2 \[\vec{q}\] + 6 \[\vec{r}\] − 9 \[\vec{s}\] \[\vec{0}\], are coplanar. Also, find the position vector of the point of intersection of the line segments PR and QS.

Answer 1

Let the point of intersection of the line segments PR  and  QS  is A. Then \[5 \vec{p} - 2 \vec{q} + 6 \vec{r} - 9 \vec{s} = \vec{0} . \]
\[ \Rightarrow 5 \vec{p} + 6 \vec{r} = 2 \vec{q} + 9 \vec{s} \]
the sum of the coefficients on both the sides of the above equation is 11 . 
So, we divide the given equation with 11 . 
\[ \Rightarrow \frac{5 \vec{p} + 6 \vec{r}}{11} = \frac{2 \vec{q} + 9 \vec{s}}{11}\]
\[\frac{5 \vec{p} + 6 \vec{r}}{5 + 6} = \frac{2 \vec{q} + 9 \vec{s}}{2 + 9}\]

Therefore, A divides PR in the ratio of 5: 6 and QS  in the ratio of \[2: 9\] 

The position vector of the point of intersection of the line segment is \[\frac{5 \vec{p} + 6 \vec{r}}{11}, \frac{2 \vec{q} + 9 \vec{s}}{11}\]