Let r- be the only point of intersection of the planes r-a-P1-r-b-P2-r-c-P3 then A- -a-b-c-0B- r - P 1 b - c - P 2 c - a - P 3 a - b C- a- b- c- are non-coplanarD- r - a b c - P 1 b - c - P 2 c - a - P 3 a - b

The correct options are C a⃗, b⃗, c⃗ are non coplanar D r⃗[a⃗b⃗c⃗]=P_1(b⃗×c⃗)+P_2(c⃗×a⃗)+P_3(a⃗×b⃗)Let r⃗= λ_1(b̅×c̅) + λ_2 (c̅×a⃗)+ λ_3( a⃗×b⃗) r⃗. a⃗= λ_1[ ...

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Byju's Answer

Standard XII

Mathematics

Section Formula

Let r⃗ be the...

Question

Let $\to r$ be the only point of intersection of the planes $\to r . \to a = P_{1} . \to r . \to b = P_{2} . \to r . \to c = P_{3}$ then

[¯ a ¯ b ¯ c] = 0

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\to r = P_{1} (\to b \times \to c) + P_{2} (\to c \times \to a) + P_{3} (\to a \times \to b)

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\to a, \to b, \to c

are non-coplanar

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\to r [\to a \to b \to c] = P_{1} (\to b \times \to c) + P_{2} (\to c \times \to a) + P_{3} (\to a \times \to b)

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Solution

The correct options are
C $\to a, \to b, \to c$ are non-coplanar

D $\to r [\to a \to b \to c] = P_{1} (\to b \times \to c) + P_{2} (\to c \times \to a) + P_{3} (\to a \times \to b)$
Let $\to r = λ_{1} (¯ b \times ¯ c) + λ_{2} (¯ c \times \to a) + λ_{3} (\to a \times \to b) \to r . \to a = λ_{1} [¯ a ¯ b ¯ c] \Rightarrow P_{1} = λ_{1} [¯ a ¯ b ¯ c] λ_{1} = \frac{P_{1}}{[¯ a ¯ b ¯ c]} s i m i l a r l y λ_{2} = \frac{P_{2}}{[¯ a ¯ b ¯ c]}, λ = \frac{P_{3}}{[¯ a ¯ b ¯ c]}$

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Let →r be the only point of intersection of the planes →r.→a=P1.→r.→b=P2.→r.→c=P3 then

The correct options are C →a,→b,→c are non-coplanar D →r[→a→b→c]=P1(→b×→c)+P2(→c×→a)+P3(→a×→b)Let →r=λ1(¯b×¯c)+λ2(¯c×→a)+λ3(→a×→b)→r.→a=λ1[¯a¯b¯c]⇒P1=λ1[¯a¯b¯c]λ1=P1[¯a¯b¯c] similarly λ2=P2[¯a¯b¯c], λ=P3[¯a¯b¯c]

Let $\to r$ be the only point of intersection of the planes $\to r . \to a = P_{1} . \to r . \to b = P_{2} . \to r . \to c = P_{3}$ then