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Rotation Concept

If z1, z2, z3...

Question

If $z_{1}, z_{2}, z_{3}$ are the vertices of an equilateral $△ A B C$ such that $| z_{1} - i | = | z_{2} - i | = | z_{3} - i |$ , then $| z_{1} + z_{2} + z_{3} | =$

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Solution

Given that
$| z_{1} - i | = | z_{2} - i | = | z_{3} - i |$
Let $A (z_{1}), B (z_{2}), C (z_{3})$
here $z_{1}, z_{2}, z_{3}$ are equidistanced from point $a = 0 + i$
$∴ a$ will be centriod of $△ A B C$
as $△ A B C$ is equilateral triangle
$∴ \frac{z_{1} + z_{2} + z_{3}}{3} = i$
$\Rightarrow | z_{1} + z_{2} + z_{3} | = 3$

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Rotation Concept

Standard XII Mathematics

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