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Vector Triple Product

If a1, b1, c1...

Question

If $^{1},^{1},^{1}$ is the reciprocal system of vector triad of $\to a, \to b, \to c$ , then $(\to a + \to b) \cdot^{1} + (\to b + \to c) \cdot^{1} + (\to c + \to a) \cdot^{1} =$

A

0

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B

1

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C

2

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D

3

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Solution

The correct option is D $3$
We know, for the reciprocal system, $^{1} = \frac{\to b \times \to c}{[\to a \to b \to c]}$
Similarly, $^{1} = \frac{\to c \times \to a}{[\to a \to b \to c]}$
and $^{1} = \frac{\to a \times \to b}{[\to a \to b \to c]}$
Hence,
$(\to a + \to b) \cdot^{1} + (\to b + \to c) \cdot^{1} + (\to c + \to a) \cdot^{1} = Σ (\to a + \to b) \cdot \frac{\to b \times \to c}{[\to a \to b \to c]} = \sum \frac{\to a \cdot (\to b \times \to c) + \to b \cdot (\to b \times \to c)}{[\to a \to b \to c]} = (1 + 0) + (1 + 0) + (1 + 0) = 3$

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\to a =^i +^j +^k, \to b = 2^i +^k, \to c =^i + 2^j + 3^k

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Vector Triple Product

Standard XII Mathematics

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