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Question
In triangle ABC, which of the following is not true:
Options
`vec(AB) + vec(BC) + vec(CA) = vec 0`
`vec(AB) + vec(BC) - vec(AC) = vec 0`
`vec(AB) + vec(BC) - vec(AC) = vec 0`
`vec(AB) - vec(CB) + vec(CA) = vec 0`
Solution
`vec(AB) + vec(BC) - vec(AC) = vec 0`
Explanation:
On applying the triangle law of addition in the given triangle, we have:
`vec(AB) + vec(BC) = vec(AC)` ......(1)
⇒ `vec(AB) + vec(BC) = -vec(CA)`
⇒ `vec(AB) + vec(BC) + vec(CA) = vec0` .....(2)
∴ The equation given in alternative A is true.
`vec(AB) + vec(BC) = vec(AC)`
⇒ `vec(AB) + vec(BC) - vec(AC) = vec0`
∴ The equation given in alternative B is true,
From equation (2), we have:
`vec(AB) - vec(CB) + vec(CA) = 0`
∴The equation given in the alternative D is true.
Now, consider the equation given in alternative C:
`vec(AB) + vec(BC) - vec(CA) = vec0`
⇒ `vec(AB) + vec(BC) = vec(CA)` ....(3)
From equation (1) and (3), we have:
`vec(AC) = vec(CA)`
⇒ `vec(AC) = -vec(AC)`
⇒ `vec(AC) + vec(AC) = 0`
⇒ `2vec(AC) = vec0`
⇒ `vec(AC) = vec0` which is not true.
Hence, the equation given in alternative C is incorrect.
The correct answer is C.
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