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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.
Solution
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 \[2: 9\]
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