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Manipal Entrance Test 2013 Math Paper

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Question : 24 of 80

Marks: +1, -0

If

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

, then a isequal to

i
k
j
(i + j + k)

Solution:

Let

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

then,

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

and

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

= x + y + z

Given that

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

∵

x = x + y = x + y + z

Now ,

x = x + y \Rightarrow y = 0

and

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

Hence; x = 1

∴ a = i

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