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Angle between Two Lines

If vectors a,...

Question

If vectors $\to a, \to b$ and $\to c$ are unit vectors such that $\to a + \to b + \to c = 0$ then the value of $(\to a . \to b + \to b . \to c + \to c . \to a)$ is

A

\frac{1}{2}

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B

\frac{- 1}{2}

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C

\frac{3}{2}

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D

\frac{- 3}{2}

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Solution

The correct option is D $\frac{- 3}{2}$
Given $\to a, \to b$ and $\to c$ are three unit vectors $| \to a | = | \to b | = | \to c | = 1$
We know $\to a + \to b + \to c = 0$
Squaring both sides,
$(\to a + \to b + \to c)^{2} = 0$
$| \to a |^{2} + | \to b |^{2} + | \to c |^{2} + 2 (\to a . \to b + \to b . \to c + \to c . \to a)$
$1 + 1 + 1 + 2 (\to a . \to b + \to b . \to c + \to c . \to a) = 0$
$2 (\to a . \to b + \to b . \to c + \to c . \to a) = - 3$
$(\to a . \to b + \to b . \to c + \to c . \to a) = \frac{- 3}{2}$

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Angle between Two Lines

Standard XII Mathematics

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