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Applications of Cross Product

In triangle ...

Question

In triangle $A B C$ , which of the following is not true?

\to A B + \to B C + \to C A = \to 0

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\to A B + \to B C - \to A C = \to 0

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\to A B + \to B C - \to C A = \to 0

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\to A B - \to C B + \to C A = \to 0

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Solution

The correct option is B $\to A B + \to B C - \to C A = \to 0$
On applying the triangle law of addition in the given triangle, we have:
$\to A B + \to B C = \to A C . . . . . . . . . . (1)$
$\Rightarrow \to A B + \to B C = - \to C A$
$\Rightarrow \to A B + \to B C + \to C A = \to 0 . . . . . . . . . . (2)$
$∴$ The equation given in alternative $A$ is true.
$\to A B + \to B C = \to A C$
$\Rightarrow \to A B + \to B C - \to A C = \to 0$
$∴$ The equation given in alternative $B$ is true.
From equation $(2)$ , we have:
$\to A B - \to C B + \to C A = 0$
$∴$ The equation given in alternative $D$ is true.
Now, consider the equation given in alternative $C$ :
$\to A B + \to B C - \to C A = \to 0$
$\Rightarrow \to A B + \to B C = \to C A . . . . . . . . . (3)$
From equations $(1)$ and $(3)$ , we have:
$\to A C = \to C A$
$\Rightarrow \to A C = - \to A C$
$\Rightarrow \to A C + \to A C = \to 0$
$\Rightarrow 2 \to A C = \to 0$
$\Rightarrow \to A C = \to 0$ , which is not true.
Hence, the equation given in alternative $C$ is incorrect.
The correct answer is $C$ .

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