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Applications of Dot Product

If a̅, b̅, ...

Question

If $¯ a, ¯ b, ¯ c,$ are unit vector such that $¯ a + ¯ b + ¯ c = ¯ 0$ , then the $¯ a . ¯ b + ¯ b . ¯ c + ¯ c . ¯ a =$

A

\frac{3}{2}

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B

- \frac{3}{2}

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C

\frac{1}{2}

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D

- \frac{1}{2}

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Solution

The correct option is B $- \frac{3}{2}$
$\begin{matrix} a, b, c, a r e u n i t V e c t o r s u c h t h a t \to a + \to b + \to c = 0 (g i v e n) T h e n, \Rightarrow {(\to a + \to b + \to c)}^{2} = {∣ ∣ \to a ∣ ∣}^{2} + {∣ ∣ ∣ \to b ∣ ∣ ∣}^{2} + {∣ ∣ \to c ∣ ∣}^{2} + 2 (\to a . \to b + \to b . \to c + \to c . \to a) \Rightarrow 0 = 1 + 1 + 1 + 2 (\to a . \to b + \to b . \to c + \to c . \to a) . \frac{- 3}{2} = \to a . \to b + \to b . \to c + \to c . \to a A n s . \end{matrix}$

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Similar questions

¯ a, ¯ b, ¯ c

are unit vectors such that

¯ a + ¯ b + ¯ c = ¯ 0

¯ a . ¯ b + ¯ b . ¯ c + ¯ c . ¯ a =

| \begin{matrix} ¯ a \end{matrix} | = 5, | \begin{matrix} ¯ b \end{matrix} | = 4, | \begin{matrix} ¯ c \end{matrix} | = 3

, then the value of

| \begin{matrix} ¯ a . ¯ b + ¯ b . ¯ c + ¯ c . ¯ a \end{matrix} |

¯ a + ¯ b + ¯ c = 0

¯ a, ¯ b, ¯ c

are three vectors such that

¯ a + ¯ b + ¯ c = 0, | \begin{matrix} ¯ a \end{matrix} | = 1, | \begin{matrix} ¯ b \end{matrix} | = 2, | \begin{matrix} ¯ c \end{matrix} | = 3

¯ a \cdot ¯ b + ¯ b \cdot ¯ c + ¯ c \cdot ¯ a

¯ a

¯ b

¯ c

are unit vectors, then

{∣ ∣ ¯ a - ¯ b ∣ ∣}^{2} + {∣ ∣ ¯ b - ¯ c ∣ ∣}^{2} + {| ¯ c - ¯ a |}^{2}

does not exceed

¯ a, ¯ b, ¯ c

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¯ a, ¯ b, ¯ c

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