In a quadrilateral \(\mathrm{PQRS}, \mathrm{M}\) and \(\mathrm{N}\) are mid-points of the sides \(P Q\) and RS respectively. If \(\overline{\mathrm{PS}}+\overline{\mathrm{QR}}=\mathrm{t} \overline{\mathrm{MN}}\), then \(\mathrm{t}=\)

\(\mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{s}, \mathrm{m}, \mathrm{n}\) be the position vectors of points \(\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}, \mathrm{M}, \mathrm{N}\).
Now \(\overline{\mathrm{PS}}+\overline{\mathrm{QR}}\)
\(
=\overline{\mathrm{s}}-\overline{\mathrm{p}}+\overline{\mathrm{r}}-\overline{\mathrm{q}}=(\overline{\mathrm{s}}+\overline{\mathrm{r}})-(\overline{\mathrm{p}}+\overline{\mathrm{q}})
\)
Since \(\mathrm{M}\) and \(\mathrm{N}\) are mid-points of PQ and RS respectively, we write
\(\overline{\mathrm{m}}=\frac{\overline{\mathrm{p}}+\overline{\mathrm{q}}}{2}\) and \(\overline{\mathrm{n}}=\frac{\overline{\mathrm{r}}+\overline{\mathrm{s}}}{2}\)
\(\therefore\) eq. (1) becomes
\(
\overline{\mathrm{PS}}+\overline{\mathrm{QR}}=2 \overline{\mathrm{n}}-2 \overline{\mathrm{m}}=2(\overline{\mathrm{n}}-\overline{\mathrm{m}})=2 \overline{\mathrm{MN}}
\)
From given data, \(\mathrm{t}=2\)