Now, $$\overrightarrow {OX} = {{\overrightarrow {QR} } \over {QR}}$$

and $$\overrightarrow {OY} = {{\overrightarrow {RP} } \over {RP}}$$

Therefore, $$(\overrightarrow {OX} \times \overrightarrow {OY} ) = {{\overrightarrow {QR} } \over {QR}} \times {{\overrightarrow {RP} } \over {RP}} = {{\overrightarrow {QR} \times \overrightarrow {RP} } \over {PQ}}$$

$$ = {{PQ\sin R} \over {PQ}} = \sin R = \sin (\pi - (P + Q) = \sin (P + Q))$$

cos(P + Q) + cos(Q + R) + cos(R + P)

= $$-$$ (cosR + cosP + cosQ)

Max. of cosP + cosQ + cosR = $${3 \over 2}$$

Min. of cos(P + Q) + cos(Q + R) + cos(R + P) is = $$ - {3 \over 2}$$

Let the equation of plane be ax + by + cz = 1. Then

a + b + c = 1

2a + b $$-$$ 2c = 0

3a $$-$$ 6b $$-$$ 2c = 0

$$ \Rightarrow $$ a = 7b

c = $${{15b} \over 2}$$

b = $${{2} \over 31}$$, a = $${{14} \over 31}$$, c = $${{15} \over 31}$$

$$ \therefore $$ 14x + 2y + 15z = 31

$$\overrightarrow{OP}$$ . $$\overrightarrow{OQ}$$ + $$\overrightarrow{OR}$$ . $$\overrightarrow{OS}$$ = $$\overrightarrow{OR}$$ . $$\overrightarrow{OP}$$ + $$\overrightarrow{OQ}$$ . $$\overrightarrow{OS}$$

$$ \Rightarrow $$ $$\overrightarrow{OP}$$($$\overrightarrow{OQ}$$ $$-$$ $$\overrightarrow{OR}$$) + $$\overrightarrow{OS}$$($$\overrightarrow{OR}$$ $$-$$ $$\overrightarrow{OQ}$$) = 0

$$ \Rightarrow $$ ($$\overrightarrow{OP}$$ $$-$$ $$\overrightarrow{OS}$$)($$\overrightarrow{OQ}$$ $$-$$ $$\overrightarrow{OR}$$) = 0

$$ \Rightarrow $$ $$\overrightarrow{SP}$$ . $$\overrightarrow{RQ}$$ = 0

Similarly $$\overrightarrow{SR}$$ . $$\overrightarrow{PQ}$$ = 0 and $$\overrightarrow{SQ}$$ . $$\overrightarrow{PR}$$ = 0

$$ \therefore \overrightarrow{S}$$ is orthocentre.