How does length of vector u+v squared equal (u+v) * (u+v)

Question

For the proof of Theorem Of Pythagoras the above is given. Can anyone explain how the first step is derived? How is the length of vector u and vector v squared produce the following on the right hand side?

Answer 1

The dot product of two vectors is given by $$ v\cdot w = |v||w|\cos(\theta) $$ where $\theta$ is the angle between them and $|v|$ means length of $v$. When the two vectors are the same, the angle is $0$, so the formula reduces to $$ v\cdot v = |v|^2.$$

Another (equivalent) formula for the dot product is in terms of components: $$ v\cdot w = v_1w_1+v_2w_2+\ldots +v_nw_n $$

Again, when we go to the case where $v = w,$ it reduces to $$ v\cdot v = v_1^2 + v_2^2 + \ldots +v_n^2 = |v|^2.$$

Answer 2

That is true by definition of the Euclidean norm.

$$\|x\| = \sqrt{x_1^2 + x_2^2 + ...+x_n^2}=\sqrt{x.x}$$