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

Let a, b, c, d be the position vectors of points P , Q , R and S respectively. Then, P Q×R S=(b a) ×(d c)=b×d b×c+d×a c×a Q R×P S=(c b) ×(d a)=c×d c×a b×d a×b R ...

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Standard XII

Mathematics

Multiplication of a Vector by a Scalar

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

P Q R

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Q R S

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P R S

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P Q S

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Solution

The correct option is B $Q R S$
Let $\to a, \to b, \to c, \to d$ be the position vectors of points $P, Q, R$ and $S$ respectively.
Then, $- - \to P Q \times - - \to R S = (\to b - \to a) \times (\to d - \to c) = \to b \times \to d - \to b \times \to c + \to d \times \to a - \to c \times \to a - - \to Q R \times - \to P S = (\to c - \to b) \times (\to d - \to a) = \to c \times \to d - \to c \times \to a - \to b \times \to d - \to a \times \to b - - \to R P \times - - \to Q S = (\to a - \to c) \times (\to d - \to b) = \to a \times \to d - \to a \times \to b - \to c \times \to d - \to b \times \to c - - \to P Q \times - - \to R S - - - \to Q R \times - \to P S + - - \to R P \times - - \to Q S = \to b \times \to d - \to b \times \to c + \to d \times \to a - \to c \times \to a - \to c \times \to d + \to c \times \to a + \to b \times \to d + \to a \times \to b + \to a \times \to d - \to a \times \to b - \to c \times \to d - \to b \times \to c = 2 (\to b \times \to d) - 2 (\to b \times \to c) - 2 (\to c \times \to d) = 2 {\to b \times \to d + \to c \times \to b + \to d \times \to c}$
$= 2 (- - \to Q S \times - - \to Q R) = 4$ area of $△ Q S R$

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Q. If

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\to p, \to q

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\to p = \frac{\to b \times \to c}{[\to a \to b \to c]}, \to q = \frac{\to c \times \to a}{[\to a \to b \to c]}

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Q. Let

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Multiplication of a Vector by a Scalar

Standard XII Mathematics

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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