If - b- and - are unit vectors satisfying - - b-2 - -b- - -2 - - - -2 - 9- then -2 - - 5 b- - 5- is

The correct option is C 3 |â b̂|^2+|b̂ ĉ|^2+|ĉ â|^2=9 ⇒ 2|â|^2+2|b̂|^2+2| c|^2 2(â·b̂+b̂·ĉ+ĉ·â)=9 (Using |a⃗ b⃗|^2=|a⃗|^2+|b⃗|^2 2a⃗·b⃗) ...

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Byju's Answer

Standard XII

Mathematics

Applications of Dot Product

If â, b̂ ...

Question

If $^a$ , $^b$ and $^c$ are unit vectors satisfying $|^a -^b |^{2} + |^b -^c |^{2} + |^c -^a |^{2} = 9$ , then $| 2^a + 5^b + 5^c |$ is

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Solution

The correct option is C $3$
$∣^a -^b ∣^{2} + ∣^b -^c ∣^{2} + ∣^c -^a ∣^{2} = 9$

$\Rightarrow 2 ∣^a ∣^{2} + 2 ∣^b ∣^{2} + 2 ∣ c ∣^{2} - 2 (^a \cdot^b +^b \cdot^c +^c \cdot^a) = 9$

(Using $∣ \to a - \to b ∣^{2} =∣ \to a ∣^{2} + ∣ \to b ∣^{2} - 2 \to a \cdot \to b)$

$\Rightarrow 6 - 2 (^a \cdot^b +^b \cdot^c +^c \cdot^a) = 9$

$\Rightarrow^a \cdot^b +^b \cdot^c +^c \cdot^a = \frac{- 3}{2}$

Now, $∣^a +^b +^c ∣^{2} =∣^a ∣^{2} + ∣^b ∣^{2} + ∣^c ∣^{2} + 2 (^a \cdot^b +^b \cdot^c +^c \cdot^a)$

$\Rightarrow∣^a +^b +^c ∣^{2} = 3 + 2 \times \frac{- 3}{2}$

$\Rightarrow∣^a +^b +^c ∣^{2} = 0$

$^a +^b +^c = 0$

$\Rightarrow^a = - (^b +^c)$

$\Rightarrow∣ 2^a + 5 (^b +^c) ∣=∣ 2^a + 5 (-^a) ∣$

$=∣ - 3^a ∣$

$= 3 ∣^a ∣$

$= 3$ .

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If ^a, ^b and ^c are unit vectors satisfying |^a−^b|2+|^b−^c|2+|^c−^a|2=9, then |2^a+5^b+5^c| is

If $^a$ , $^b$ and $^c$ are unit vectors satisfying $|^a -^b |^{2} + |^b -^c |^{2} + |^c -^a |^{2} = 9$ , then $| 2^a + 5^b + 5^c |$ is