Question

Two non zero vectors $\overrightarrow{v_1}\mathrm{and}\overrightarrow{v_2}$ which are orthogonal to each other will always be linearly

• A. dependent
• B. independent
• C. $\mathrm{either}\mathrm{dependent}\mathrm{or}\mathrm{independent}$

Answer 1

The correct option is B independent

Let the linear combination be
$c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}=0$
where $c_1\mathrm{and}{c}_2\mathrm{are}\mathrm{scalars}.$

For linearly independency, sufficient condition to satisfy is
$c_1=c_2=0$
taking dot product with $\overrightarrow{v_1}$
we get
$\overrightarrow{v_1}.(c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2})=\overrightarrow{v_1}.0=0\Rightarrow c_1\overrightarrow{v_1}.\overrightarrow{v_1}+c_2\overrightarrow{v_1}.\overrightarrow{v_2}=0\mathrm{Since}\mathrm{we}\mathrm{know}\overrightarrow{v_1}.\overrightarrow{v_2}=0\Rightarrow c_1{\left(\overrightarrow{v_1}\right|}^2=0\mathrm{since}{\left(\overrightarrow{v_1}\right|}^2\mathrm{cannot}\mathrm{be}\mathrm{zero}\mathrm{So}{c}_1=0\mathrm{Similary}\mathrm{repeating}\mathrm{same}\mathrm{procedure}\mathrm{for}{c}_2\mathrm{we}\mathrm{get}{c}_2=0\mathrm{So}\mathrm{they}\mathrm{will}\mathrm{be}\mathrm{linearly}\mathrm{independent}.$