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Equation of the Circle Using End Points of Diameter

Let a⃗ and ...

Question

Let $\to a$ and $\to b$ be the position vectors of the points $A$ and $B$ . Prove that the position vectors of the points which trisects the line segment $A B$ are $\frac{\to a + 2 \to b}{3}$ and $\frac{\to b + 2 \to 2}{3}$

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Solution

$\to x$ & $\to y$ are points that trisects the line between $\to a$ & $\to b$
$| \to x - \to a | =$ $| \to y - \to x | =$ $| \to b - \to y | = \frac{1}{3} | \to b - \to a |$
$\to x - \to a = \frac{1}{3} (\to b - \to a)$
$\to x = \frac{\to b + 2 \to a}{3}$
$\to b - \to y = \frac{1}{3} (\to b - \to a) .$
$\to y = \frac{\to a + 2 \to b}{3}$

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