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Suppose $x, v$ are vectors in $\mathbb{R}^n$ such that for every choice of scalars, $c, d \in \mathbb{R}$, the vectors $(cx + dv)$ and $(dx + cv)$ are orthogonal. Show that $x = \overrightarrow{0} = v$.

I used the simple definition and got to:

$$cd(||x||^2 + ||v||^2) + x \cdot v (c^2 + d^2) = 0$$

But how do I move on?

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    First, remember that $c,d$ above are variables (you can choose them freely), while $x,v$ are constants (given). Then, can you maybe find a term above that takes a special value if and only if $x=v=\overrightarrow{0}$? – 2017-02-06

1 Answers 1

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For $c=1$ and $d=0$ we get: $x \cdot v=0$

F0r $c=d=1$ we get: $0=(x+v) \cdot (x+v)= ||x||^2+2x \cdot v+||v||^2=||x||^2+||v||^2$.

Hence $x=0=v$.