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Properties of Set Operation

If a⃗=b⃗+c⃗...

Question

If $\to a = \to b + \to c$ , then is it true that $| \to a | = | \to b | + | \to c |$ ? Justify your answer.

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Solution

In $Δ A B C$ , let $\to C B = \to a, \to C A = \to b$ , and $\to A B = \to c$ (as shown in the following figure).
Now, by the triangle law of vector addition, we have $\to a = \to b + \to c$ .
It is clearly known that $| \to a |, | \to b |$ , and $| \to c |$ represent the sides of $Δ A B C$ .
Also, it is known that the sum of the lengths of any two sides of a triangle is greater than the third side.
$∴ | \to a | < | \to b | + | \to c |$
Hence, it is not true that $| \to a | = | \to b | + | \to c |$ .

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