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Addition of Vectors

If a. b, c ...

Question

If $¯ ¯ ¯ a . ¯ ¯ b, ¯ ¯ c$ are unit vectors, then $| ¯ ¯ ¯ a - ¯ ¯ b |^{2} + | ¯ ¯ b - ¯ ¯ c |^{2} + | ¯ ¯ c - ¯ ¯ ¯ a |^{2}$ does not exceed.

A

4

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B

9

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C

8

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D

6

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Solution

The correct option is B $9$
$\begin{matrix} | \to a | = ∣ ∣ \to b ∣ ∣ = | \to c | = 1 {∣ ∣ \to a - \to b ∣ ∣}^{2} + {∣ ∣ \to b - \to c ∣ ∣}^{2} + {| \to c - \to a |}^{2} = 2 ({| a |}^{2} + {| b |}^{2} + {| c |}^{2}) - 2 (\to a \cdot \to b + \to b \cdot \to c + \to c \cdot \to a) ∣ ∣ \to a + \to b + \to c ∣ ∣ \geq 0 {∣ ∣ \to a + \to b + \to c ∣ ∣}^{2} \geq 0 {| a |}^{2} + {| b |}^{2} + {| c |}^{2} + 2 (\to a \cdot \to b + \to b \cdot \to c + \to c \cdot \to a) \geq 0 \to a \cdot \to b + \to b \cdot \to c + \to c \cdot \to a \geq - \frac{3}{2} 2 (\to a \cdot \to b + \to b \cdot \to c + \to c \cdot \to a) \geq - 3 {∣ ∣ \to a - \to b ∣ ∣}^{2} + {∣ ∣ \to b - \to c ∣ ∣}^{2} + {| \to c - \to a |}^{2} \geq 2 \times 3 + 3 \geq 9 H e n c e, t h e c o r r e c t o p t i o n i s (B) \end{matrix}$

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