If a-b- and c- are three unit vectors such that a-b-c- is also a unit vector and -1- -2 and -3 are the angles between the vectors a-b- b- c- and c- a- then

The correct option is C At least one is obtuse angleSince #160; | a⃗ +b⃗ +c⃗ #160;| =1⇒( a⃗ +b⃗ +c⃗ #160;) .( a⃗ +b⃗ +c⃗ #160;) =1 ⇒ 1+1+1+2( ...

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Standard XII

Mathematics

Addition of Vectors

If a⃗,b⃗ an...

Question

If $\to a, \to b$ and $\to c$ are three unit vectors such that $\to a + \to b + \to c$ is also a unit vector and $θ_{1}, θ_{2}$ and $θ_{3}$ are the angles between the vectors $\to a, \to b;$ $\to b, \to c$ and $\to c, \to a$ , then

All are acture angles

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All are right angles

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At least one is obtuse angle

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Exactly one is obtuse angle

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Solution

The correct option is C At least one is obtuse angle
Since $∣ ∣ \to a + \to b + \to c ∣ ∣ = 1 \Rightarrow (\to a + \to b + \to c) . (\to a + \to b + \to c) = 1$
$\Rightarrow 1 + 1 + 1 + 2 (\to a . \to b + \to b . \to c + \to c . \to a) = 1$
$(\to a . \to b + \to b . \to c + \to c . \to a) = - 1$
$cos θ_{1} + cos θ_{2} + cos θ_{3} = - 1$
So atleast one of these $3$ values $cos θ$ must be negative.
So atleast one is obtuse angle.

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If →a,→b and →c are three unit vectors such that →a+→b+→c is also a unit vector and θ1, θ2 and θ3 are the angles between the vectors →a,→b; →b, →c and →c, →a, then

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