State True and False
If $z_{1} = 1 + 2 i, z_{2} = 2 + 3 i, z_{3} = 3 + 4 i,$ then $z_{1}, z_{2}$ and $z_{3}$ are collinear.
Type 1 for true and 0 for false

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Solution

The three points are collinear as $2 z_{2} = z_{1} + z_{3}$

By mid-point formula, $z_{2}$ is the midpoint of $z_{1}$ and $z_{3}$

And hence, all the points $z_{1}, z_{2} a n d z_{3}$ are co-linear.
True

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Similar questions

z_{1} = 1 + 2 i, z_{2} = 2 + 3 i, z_{3} = 3 + 4 i

z_{1}, z_{2}

z_{3}

z_{1} = 1 + 2 i, z_{2} = 2 + 3 i, z_{3} = 3 + 4 i,

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z_{3}

z_{1} = 1 + 2 i, z_{2} = 2 + 3 i, z_{3} = 3 + 4 i, p r o v e t h a t t h e n z_{1}, z_{2} a n d z_{3}

z_{1}, z_{2}, z_{3}

\frac{p}{| z_{2} - z_{3} |} = \frac{q}{| z_{3} - z_{1} |} = \frac{r}{| z_{1} - z_{2} |}

\frac{p^{2}}{z_{2} - z_{3}} + \frac{q^{2}}{z_{3} - z_{1}} + \frac{r^{2}}{z_{1} - z_{2}} = 0

z_{1} + z_{2} + z_{3} = A, z_{1} + z_{2} ω + z_{3} ω^{2} = B,

z_{1} + z_{2} ω^{2} + z_{3} ω = C .

A + B + C = 3 z_{1}

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