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

|a b|^2+|b c|^2+|c a|^2=9 ⇒ |a|^2+|b|^2 2a· nbsp;b+|b|^2+|c|^2 2b·c+|c|^2+|a|^2 2a· nbsp;c=9 ⇒a·b+b·c+c·a= 32 Now, |a+b+c|^2=|a|^2+|b|^2+|c|^2+2a·b+2b·c+2a ...

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Byju's Answer

Standard XII

Mathematics

Condition for Two Lines to be on the Same Plane

If a, b and c...

Question

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

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Solution

$| \to a - \to b |^{2} + | \to b - \to c |^{2} + | \to c - \to a |^{2} = 9 \Rightarrow | \to a |^{2} + | \to b |^{2} - 2 \to a \cdot \to b + | \to b |^{2} + | \to c |^{2} - 2 \to b \cdot \to c + | \to c |^{2} + | \to a |^{2} - 2 \to a \cdot \to c = 9$
$\Rightarrow \to a \cdot \to b + \to b \cdot \to c + \to c \cdot \to a = - \frac{3}{2}$

Now,
$| \to a + \to b + \to c |^{2} = | \to a |^{2} + | \to b |^{2} + | \to c |^{2} + 2 \to a \cdot \to b + 2 \to b \cdot \to c + 2 \to a \cdot \to c$
$\Rightarrow | \to a + \to b + \to c |^{2} = 3 + 2 (- \frac{3}{2})$
$\Rightarrow | \to a + \to b + \to c |^{2} = 0$
$\Rightarrow \to a + \to b + \to c = 0 \Rightarrow \to b + \to c = - \to a \dots (1)$
Using $(1)$ , we get
$| 2 \to a + 5 \to b + 5 \to c | = | 2 \to a - 5 \to a | = 3$

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