Question

For any four points P, Q, R, S,

$\left| \overset{\rightarrow}{PQ} \times \overset{\rightarrow}{RS}–\overset{\rightarrow}{QR} \times \overset{\rightarrow}{PS} + \overset{\rightarrow}{RP} \times \overset{\rightarrow}{QS} \right|$is equal to

4 times the area of the triangle –

• A. PQR
• B. QRS
• C. PRS
• D. PQS

Answer 1

Answer: (B)

QRS

Answer 2

$\overset{\rightarrow}{PQ}$× $\overset{\rightarrow}{RS}$ –$\overset{\rightarrow}{QR}$ × $\overset{\rightarrow}{PS}$ + $\overset{\rightarrow}{RP}$ × $\overset{\rightarrow}{QS}$

= ($\overset{\rightarrow}{PS}$+ $\overset{\rightarrow}{SQ}$) ×$\overset{\rightarrow}{RS}$– $\overset{\rightarrow}{QR}$× $\overset{\rightarrow}{PS}$ + ($\overset{\rightarrow}{RS}$ + $\overset{\rightarrow}{SP}$ ) × $\overset{\rightarrow}{QS}$

= $\overset{\rightarrow}{PS}$ × ($\overset{\rightarrow}{RS}$+$S\overset{\rightarrow}{Q}$ + $\overset{\rightarrow}{QR}$) – $\overset{\rightarrow}{SQ}$× $\overset{\rightarrow}{SR}$– $\overset{\rightarrow}{SQ}$× $\overset{\rightarrow}{SR}$

= $\overset{\rightarrow}{PS}$ ×$\overset{\rightarrow}{0}$ – 2 $\overset{\rightarrow}{SQ}$ × $\overset{\rightarrow}{SR}$ = 4 × area of DQRS