Okay, so I'm reading a proof which shows that the dot product of two perpendicular vectors is 0.
I am kinda confused by the beginning of the proof, the part where it says that for two perpendicular vectors v and w, the hypotenuse is v-w. Why is this so? Am I drawing the sketches for addition and subtraction incorrectly?
The Figure 1.7 shows that the hypotenuse is the addition of the two vectors, but the proof says that it's the subtraction.
Of course, if the hypotenuse is represented as v+b the proof wouldn't stand
What am I doing wrong?
Best Answer
Your sketch shows perfectly that the hypotenuse is what the author claims it is. The problem is that you are on a vector space, so every vector is pinned to the origin (otherwise you would be working on an affine space). This may be the confusion point. When the author says that the third side is $v-w$, he or she means that this side corresponds to the vector $v-w$ (which is necessarily pinned at the origin) translated to some position in the affine plane.
The hypotenuse of the right triangle defined by $v$ and $w$ has in any case the same direction as $v-w$ (and not the same direction as $v+w$). Note that $w-v$ would also work, which is something you should expect (since the roles of $v$ and $w$ are symmetric).