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CGPET 2019 Solved Paper

Section: Mathematics

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Question : 101 of 150

Marks: +1, -0

Mathematics

(r \cdot \hat{i})^{2}+(r \cdot \hat{j})^{2}+(r \cdot \hat{k})^{2}

0
1
$3 r^{2}$
$r^{2}$

Solution:

Let

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

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

(r \cdot \hat{i})=x \Rightarrow (r \cdot \hat{i})^{2}=x^{2}

.......(i)

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

(r \cdot \hat{j})=y \Rightarrow (r \cdot \hat{j})^{2}=y^{2}

...........(ii)

And

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

(r \cdot \hat{k})=z \Rightarrow (r \cdot \hat{k})^{2}=z^{2}

.........(iii)

Now,

(r \cdot \hat{i})^{2}+(r \cdot \hat{j})^{2}+(r \cdot \hat{k})^{2}

Put the value of Eqs.(i), (ii) and (iii), we get

=x^{2}+y^{2}+z^{2}=r^{2}

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