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प्रश्न
If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is `sqrt(3)`.
उत्तर
Let three unit vectors are a, b and c
given that the sum of the unit vectors is a unit vector.
∴ a + b = c
or | c |2 = | a + b |2
or | c |2 = | a |2 + | b |2 + 2| a | | b |cos θ
or 1 = 1 + 1 + 2 cos θ ...[∵ | a | = | b | = | c | = 1 (unit vector)]
⇒ `cos θ = -1/2` ...(1)
Now, | a - b |2 =| a |2 + | b |2 - 2| a | | b |cos θ
| a - b |2 = [ 1 + 1 + 1 ]
`|a -b|= sqrt(3)`
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