For any four points P- Q- R- S- -P-Q-R-S-Q-R-P-S-R-P-Q-S- is equal to 4 times the area of the triangle-

The correct option is B QRSConsider the problem #160;Let the points P,Q,R,S #160;According to question #160; PQ #160; ×RS #160; QR #160 ...

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Standard XII

Mathematics

Applications of Cross Product

For any four ...

Question

For any four points P, Q, R, S, $∣ ∣ \to P Q \times \to R S - \to Q R \times \to P S + \to R P \times \to Q S ∣ ∣$ is equal to $4$ times the area of the triangle.

PQR

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QRS

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PRS

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PQS

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Solution

The correct option is B QRS
Consider the problem
Let the points $P, Q, R, S$
According to question
$- - \to P Q \times - - \to R S - - - \to Q R \times - \to P S + - - \to R P \times - - \to Q S = (- \to P S + - - \to S Q) \times - - \to R S - - - \to Q R \times - \to P S + (- - \to R S + - \to S P) \times - - \to Q S$

$= - \to P S \times (- - \to R S + - - \to S Q + - - \to Q R) - - - \to S Q \times - - \to S R - - - \to S Q \times - - \to S R$

$= - \to P S \times \to O - 2 - - \to S Q \times - - \to S R$

$= 4 \times a r e a o f Δ Q R S$

Hence option $B$ is correct answer.

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For any four points P, Q, R, S, ∣∣→PQ×→RS−→QR×→PS+→RP×→QS∣∣ is equal to 4 times the area of the triangle.

For any four points P, Q, R, S, $∣ ∣ \to P Q \times \to R S - \to Q R \times \to P S + \to R P \times \to Q S ∣ ∣$ is equal to $4$ times the area of the triangle.