Two non zero vectors v1- and v2- which are orthogonal to each other will always be linearly

Let the linear combination be c1v1→+c2v2→=0 nbsp;where nbsp;c1 nbsp;and nbsp;c2 nbsp;are nbsp;scalars. For linearly independency, sufficient condi ...

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Byju's Answer

Standard XII

Mathematics

Linear Dependence and Independence of Vectors

Two non zero ...

Question

Two non zero vectors $_{1} and_{2}$ which are orthogonal to each other will always be linearly

dependent

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independent

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either dependent or independent

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Solution

The correct option is B independent
Let the linear combination be
$c_{1}_{1} + c_{2}_{2} = 0$
where $c_{1} and c_{2} are scalars .$

For linearly independency, sufficient condition to satisfy is
$c_{1} = c_{2} = 0$
taking dot product with $_{1}$
we get
$_{1} . (c_{1}_{1} + c_{2}_{2}) =_{1} . 0 = 0 \Rightarrow c_{1}_{1} ._{1} + c_{2}_{1} ._{2} = 0 Since we know_{1} ._{2} = 0 \Rightarrow c_{1} {(_{1} ∣ ∣}^{2} = 0 since {(_{1} ∣ ∣}^{2} cannot be zero So c_{1} = 0 Similary repeating same procedure for c_{2} we get c_{2} = 0 So they will be linearly independent .$

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Two non zero vectors →v1 and →v2 which are orthogonal to each other will always be linearly

Two non zero vectors $_{1} and_{2}$ which are orthogonal to each other will always be linearly