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?
[Math] How does length of vector u+v squared equal (u+v) * (u+v)
analysisfunctional-analysislinear algebravectors
Related Solutions
The components axes $x^\prime$ and $y^\prime$ are inclined by angle $\theta$.
Use the sine rule.
It depends what point on the $Z$-axis r ends on. Assuming you want the shortest r possible:
r is shortest when it is perpendicular to the $Z$-axis ends
$\implies$ r ends at $(0, 0, z)$
Note: that since r goes from $(x, y, z)$ to $(0, 0, z)$ it is parallel to the $xy$-plane
$\implies$ the length of r is the same as the length $(x, y, 0)$ to $(0, 0, 0)$
So,
||r|| = $\sqrt{ x^2 + y^2}$
Best Answer
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 $$ v\cdot v = v_1^2 + v_2^2 + \ldots +v_n^2 = |v|^2.$$