If a-b andcare three vectors such that -a b c-1- then the value of -a-b b-c c-a-a-b b-c c-a- -a- b-c b- c-a c- a-b- is

Solving the equation by splitting in three parts: Part I = [a+b b+c c+a] = (a+b) [(b+c)× (c+a)] = (a+b) [b×c +b×a +c×c +c×a] = (a+b) [b×c +b×a +0 +c×a] = ...

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Standard XII

Mathematics

Scalar Triple Product

If a,b andcar...

Question

If $\to a, \to b$ and $\to c$ are three vectors such that $[\to a \to b \to c] = 1$ , then the value of $[\to a + \to b \to b + \to c \to c + \to a] + [\to a \times \to b \to b \times \to c \to c \times \to a] + [\to a \times (\to b \times \to c) \to b \times (\to c \times \to a) \to c \times (\to a \times \to b)]$ is

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Solution

The correct option is C $3$
Solving the equation by splitting in three parts:
Part - $I$
$= [\to a + \to b \to b + \to c \to c + \to a]$
$= (\to a + \to b) [(\to b + \to c) \times (\to c + \to a)]$
$= (\to a + \to b) [\to b \times \to c + \to b \times \to a + \to c \times \to c + \to c \times \to a]$
$= (\to a + \to b) [\to b \times \to c + \to b \times \to a + 0 + \to c \times \to a]$
$= [\to a \to b \to c] + [\to a \to b \to a] + [\to a \to c \to a] + [\to b \to b \to c] + [\to b \to b \to a] + [\to b \to c \to a]$
$= [\to a \to b \to c] + 0 + 0 + 0 + 0 + [\to b \to c \to a]$
$= 2 [\to a \to b \to c] = 2 (1) = 2$
Part - $I I$
$= [\to a \times \to b \to b \times \to c \to c \times \to a]$
$= (\to a \times \to b) [(\to b \times \to c) \times (\to c \times \to a)]$
$= (\to a \times \to b) {[\to b \to c \to a] \to c - [\to b \to c \to c] \to a}$
$= (\to a \times \to b) {[\to b \to c \to a] \to c - 0}$
$= (\to a \times \to b) {\to c [\to a \to b \to c]}$
$= [\to a \to b \to c] [\to a \to b \to c]$
$= [\to a \to b \to c]^{2} = 1^{2} = 1$
Part - $I I I$
$= [\to a \times (\to b \times \to c) \to b \times (\to c \times \to a) \to c \times (\to a \times \to b)]$
$= {\to a \times (\to b \times \to c)} {(\to b \times (\to c \times \to a)) \times (\to c \times (\to a \times \to b))}$
$= 0$
Adding Parts $I, I I, I I I$
$\Rightarrow 2 + 1 + 0 = 3$

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Scalar Triple Product

Standard XII Mathematics

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If →a,→b and→care three vectors such that [→a →b →c]=1, then the value of [→a+→b →b+→c →c+→a]+[→a×→b →b×→c →c×→a]+[→a×(→b×→c) →b×(→c×→a) →c×(→a×→b)] is