Question #143591
Orthogonal
Is it true that if x is orthogonal to v and w, then x is also orthogonal to v − w. Why so?
1
Expert's answer
2020-11-11T08:52:09-0500

It is well known that the scalar product is a distributive with regard to the sum and the difference of vectors:


a(b±c)=ab±aca\cdot(b\pm c)=a\cdot b\pm a\cdot c.


The vectors aa and bb are ortogonal if and only if the scalar product aba\cdot b is equal to 0.


If  xx is orthogonal to vv and ww, then xv=0x\cdot v=0 and xw=0x\cdot w=0. Therefore,


x(vw)=xvxw=00=0,x\cdot(v-w)=x\cdot v- x\cdot w =0-0=0,


and we conclude that xx is also orthogonal to vw.v − w.




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