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If the vectors $\hat{\mathrm{i}}-\mathrm{x} \hat{\mathrm{j}}-\mathrm{y} \hat{\mathrm{k}}$ and $\hat{\mathrm{i}}+\mathrm{x} \hat{\mathrm{j}}+\mathrm{y} \hat{\mathrm{k}}$ are orthogonal to
each other, then what is the locus of the point $(\mathrm{x}, \mathrm{y})$ ?
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each other, then what is the locus of the point $(\mathrm{x}, \mathrm{y})$ ?
Solution:
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Verified Answer
The correct answer is:
a circle
Since both vectors are orthogonal $\therefore$ their dot product is zero.
$\therefore 1(1)+(-x)(x)+(-y)(y)=0$
$\Rightarrow 1-x^{2}-y^{2}=0$
$\Rightarrow x^{2}+y^{2}=1$
Which is a circle.
$\therefore 1(1)+(-x)(x)+(-y)(y)=0$
$\Rightarrow 1-x^{2}-y^{2}=0$
$\Rightarrow x^{2}+y^{2}=1$
Which is a circle.
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