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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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.