A a - B b - C c are the vertices of a triangle ABC and R r is any point in the plane of triangle ABC- then r - a - b - b - c - c - a is alwaysequal toA- - a b c -B- Zero C- -a-b-c-D- None of these

The correct option is A [a⃗ b⃗ c⃗] A vector perpendicular to the plane of A (a⃗), B (b⃗), C(c⃗) is (b⃗ a⃗) × (c⃗ a⃗) =a⃗×b⃗+b⃗×c⃗+c⃗×a⃗ Now for any poin ...

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Byju's Answer

Standard IX

Mathematics

Test for Coplanarity

A a , B b , C...

Question

$A (\to a), B (\to b), C (\to c)$ are the vertices of a triangle ABC and $R (\to r)$ is any point in the plane of triangle ABC, then $\to r . (\to a \times \to b + \to b \times \to c + \to c \times \to a)$ is always equal to

Zero

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$[\to a \to b \to c]$

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$- [\to a \to b \to c]$

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None of these

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Solution

The correct option is B
$[\to a \to b \to c]$

A vector perpendicular to the plane of $A (\to a), B (\to b), C (\to c) i s (\to b - \to a) \times (\to c - \to a) = \to a \times \to b + \to b \times \to c + \to c \times \to a$
Now for any point $R (\to r)$ in the plane of A, B, C
$(\to r - \to a) . (\to a \times \to b + \to b \times \to c + \to c \times \to a) = 0 \Rightarrow \to r . (\to a \times \to b + \to b \times \to c + \to c \times \to a) - \to a . (\to a \times \to b + \to b \times \to c + \to c \times a) = 0 \Rightarrow \to r . (\to a \times \to b + \to b \times \to c + \to c \times \to a) = 0 + \to a . \to b \times \to c + 0 \Rightarrow \to r . (\to a \times \to b + \to b \times \to c + \to c \times \to a) = [\to a \to b \to c]$

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$A (\to a), B (\to b), C (\to c)$ are the vertices of a triangle ABC and $R (\to r)$ is any point in the plane of triangle ABC, then $\to r . (\to a \times \to b + \to b \times \to c + \to c \times \to a)$ is always equal to