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Condition for Coplanarity of Four Points

If a , b , ...

Question

If $\to a, \to b, \to c$ are three non-coplanar vectors, $\to p, \to q, \to r$ are non-zero vectors such that $\to p = \frac{\to b \times \to c}{[\to a \to b \to c]}, \to q = \frac{\to c \times \to a}{[\to a \to b \to c]}, \to r = \frac{\to a \times \to b}{[\to a \to b \to c]},$ then the value of the expression $(\to a + \to b + \to c), (\to p + \to q + \to r)$ is

A

0

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B

4

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C

2

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D

3

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Solution

The correct option is A $0$
$\begin{matrix} [¯ ¯ ¯ a ¯ ¯ b ¯ ¯ c] \neq 0 ¯ ¯¯ ¯ P = \frac{¯ b \times ¯ c}{[¯ ¯ ¯ a ¯ b ¯ c]}, ¯ ¯¯ ¯ Q = \frac{¯ c \times ¯ ¯ ¯ a}{[¯ ¯ ¯ a ¯ b ¯ ¯ ¯ c]} ¯ ¯¯ ¯ R = \frac{¯ ¯ ¯ a \times ¯ b}{[¯ ¯ ¯ a ¯ b ¯ c]} T o f i n d (¯ ¯ ¯ a + ¯ ¯ b + ¯ ¯ c) . (¯ ¯ ¯ p + ¯ ¯ ¯ q + ¯ ¯ ¯ r) ¯ ¯ ¯ a . ¯ ¯ ¯ p = ¯ ¯ ¯ a . \frac{(¯ b \times ¯ c)}{[¯ ¯ ¯ a ¯ b ¯ ¯ ¯ c]} = \frac{1}{[¯ ¯ ¯ a ¯ b ¯ ¯ ¯ c]} = {¯ ¯ ¯ a . (¯ ¯ b \times ¯ ¯ c)} = \frac{[¯ ¯ ¯ a ¯ b ¯ c]}{[¯ ¯ ¯ a ¯ b ¯ c]} = 1 ¯ ¯ b . ¯ ¯ ¯ p = ¯ ¯ b . \frac{(¯ b \times ¯ c)}{[¯ ¯ ¯ a ¯ b ¯ ¯ ¯ c]} = \frac{1}{[¯ ¯ ¯ a ¯ b ¯ ¯ ¯ c]} {¯ ¯ b . (¯ ¯ b \times ¯ ¯ c)} = 0 ¯ ¯ c . ¯ ¯ ¯ p = 0 ¯ ¯ b . ¯ ¯ ¯ q = 1 ¯ ¯ c . ¯ ¯ ¯ r = 1 R e m a i n i n g Z e r o . \end{matrix}$

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Similar questions

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

a, b, c

are three non-coplanar vectors then the value of the expression

(a + b + c) . (p + q + r)

\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]}

\to r = \frac{\to a \times \to b}{[\to a \to b \to c]}

\to a, \to b

\to c

are three non-coplanar vectors, then the value of the expression

(\to a + \to b + \to c) \cdot (\to p + \to q + \to r)

a, b

c

be three non-coplanar vectors and let

p, q

r

be vectors defined by the relations

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

r = \frac{a \times b}{a b c}

Then the value of the expression

(a + b) . p + (b + c) . q + (c + a) . r

\to a, \to b, \to c

be three non-coplanar vector and

\to p, \to q, \to r

be vectors defined by the relations

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

then the value of the expression

(\to a + \to b) . \to p + (\to b + \to c) . \to q + (\to c + \to a) . \to r

\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]}

\to r = \frac{\to a \times \to b}{[\to a \to b \to c]}

\to a, \to b

\to c

are three non-coplanar vectors, then the value of the expression

(\to a + \to b + \to c) \cdot (\to p + \to q + \to r)

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