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I know that magnitude of $ X$ is defined as:

$$\|X\|=\sqrt {( {X\cdot X})}$$

Now if I define $ X$ as the sum of two vector like this $ X= X_1+ X_2$

then what will be the magnitude of:

$$\| X_1+ X_2\|=?$$

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2 Answers

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The magnitude is given by the same formula as the one you gave, that is, $$\sqrt{(X_1+X_2)\cdot(X_1+X_2)}.$$

We can play around with this formula in various ways. For example, we have $$(X_1+X_2)\cdot (X_1+X_2)=X_1\cdot X_1+X_2\cdot X_2 +2 X_1\cdot X_2.$$

Note that $X_i\cdot X_i=\Vert X_i\Vert^2$. Also, $X_1\cdot X_2=\Vert X_1\Vert \Vert X_2\Vert \cos\theta,$ where $\theta$ is the angle between the two vectors. So an alternate expression for the magnitude of the sum is $$\sqrt{\Vert X_1\Vert^2+ \Vert X_2\Vert^2+ 2\Vert X_1\Vert \Vert X_2\Vert \cos\theta}.$$

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The magnitude of the vectors does not uniquely determine the magnitude of the sum. However what you can do is to use the triangle inequality to get an upper and a lower bound for the possible magnitudes.

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