Question

In triangle ABC, which of the following is not true:

विकल्प

• `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`

Answer 1

`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.