Can two perpendicular vectors to each other be linearly dependent and can two parallel vectors to each other be linearly independent ?


closed as off-topic by Namaste, user99914, egreg, Mohammad Riazi-Kermani, Brian Borchers Mar 24 '18 at 23:20

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  • 3
    $\begingroup$ 1) Yes, if one of them is zero. 2) No. $\endgroup$ – Qiaochu Yuan Jan 16 '18 at 20:29
  • $\begingroup$ So if non of two perpendicular vectors is 0 then they cant be linearly dependent right ? $\endgroup$ – Endrit Shabani Jan 16 '18 at 20:32
  • $\begingroup$ ^ Yes that's right $\endgroup$ – John Doe Jan 16 '18 at 20:32
  • 1
    $\begingroup$ @QiaochuYuan Slightly strengthening 1): if and only if. $\endgroup$ – Thomas Jan 16 '18 at 20:32
  • $\begingroup$ but isn't the 0 vector perpendicular to any vector ? $\endgroup$ – Endrit Shabani Jan 16 '18 at 20:33

Every set which contains mutually perpendicular vectors is a independent set. All the vectors in this set are independent. You can search for Gram-Schmidt process. In that process it makes an orthonormal basis for $\mathbb R^n$. There you can easily see why a set of mutually perpendicular vectors are independent.

If two vectors are parallel then each of them is a non-zero scalar multiple of other one unless one of them is zero. In both situation they form a dependent set


Note that for $v_1\neq 0$ and $v_2\neq 0$ and $v_1\cdot v_2=0$

$$av_1+bv_2=0 \iff av_1\cdot v_1+bv_1\cdot v_2=0\iff a|v_1|^2=0\iff a=0$$

$$av_1+bv_2=0 \iff av_1\cdot v_2+bv_2\cdot v_2=0\iff b|v_2|^2=0\iff b=0$$

thus $v_1$ and $v_2$ are linearly independent.


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