TITLE: What is magnitude of sum of two vector?
QUESTION [8 upvotes]: 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:
$$\| X_1+ X_2\|=?$$

REPLY [21 votes]: 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
$$\sqrt{\Vert X_1\Vert^2+ \Vert X_2\Vert^2+ 2\Vert X_1\Vert \Vert X_2\Vert \cos\theta}.$$