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KEAM 2019 Math Paper

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Question : 109 of 121

Marks: +1, -0

If

a \cdot \hat{i} = a \cdot (\hat{i} + \hat{j}) = a \cdot (\hat{i} + \hat{j} + \hat{k})

, then a is equal to

$\hat{i}$
$\hat{k}$
$\hat{j}$
$\hat{i} + \hat{j} + \hat{k}$
$\hat{j}$

Solution:

Let

a = x \hat{i} + \widehat{yj} + z \hat{k}

a \cdot \hat{i} = (x \hat{i} + y \hat{j} + z \hat{k}) \cdot \hat{i} = x

a \cdot (\hat{i} + \hat{j}) = (x \hat{i} + y \hat{j} + z \hat{k}) \cdot (\hat{i} + \hat{j})

= x + y

and

a \cdot (\hat{i} + \hat{j} + \hat{k}) = (\widehat{xi} + \widehat{yj} + z \hat{k}) \cdot (\hat{i} + \hat{j} + \hat{k})

= x + y + z

It is given that,

a \cdot \hat{i} = a \cdot (\hat{i} + \hat{j}) = a \cdot (\hat{i} + \hat{j} + \hat{k})

\Rightarrow \;\; x = x + y = x + y + z

Taking,

\;\; x = x + y \Rightarrow y = 0

x + y = x + y + z \Rightarrow z = 0

x

has any real values.

Now, take

x = 1

\therefore

a = \hat{i}

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