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Let \hat{i}=a\left ( \hat{i}+\hat{j} \right )+b\left ( 2\hat{i}-\hat{j} \right ) then a=b equals

Option: 1

-\frac{1}{3}


Option: 2

0


Option: 3

\frac{1}{3}


Option: 4

\frac{2}{3}


Answers (1)

best_answer

As we learned

Collinear Vectors -

Representation of any vector as a linear combination of non-collinear vectors \vec{a} and \vec{b} is unique.

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 \hat{i}=a(\hat{i}+\hat{j})+b(2\hat{i}-\hat{j})

It means we are trying to represent \hat{i} as linear combination of non-collinear vectors \hat{i}+\hat{j}\, \, and\, \, 2\hat{i}-\hat{j}

\therefore We will have unique values of a & b

\hat{i}=(a+2b)\hat{i}+(a-b)\hat{j}

\Rightarrow (a+2b-1)\hat{i}+(a-b)\hat{j}=\vec{0}

It is possible only when a+2b=1 and a=b

\Rightarrow a=\frac{1}{3},\, \, b=\frac{1}{3}

So, option (D)

 

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