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Co-Ordinates of a Point in a Plane

If a → and b ...

Question

If $\vec{a} a n d \vec{b}$ are the position vectors of A and B respectively, then the position vector of a point C on AB produced such that $\vec{A C} = 3 \vec{A B} is______________________.$

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Solution

Given:
$\vec{A C} = 3 \vec{A B}$

Let $\vec{c}$ is the position vectors of C.

$\vec{A C} = 3 \vec{A B} \Rightarrow \vec{c} - \vec{a} = 3 (\vec{b} - \vec{a}) \Rightarrow \vec{c} - \vec{a} = 3 \vec{b} - 3 \vec{a} \Rightarrow \vec{c} = 3 \vec{b} - 3 \vec{a} + \vec{a} \Rightarrow \vec{c} = 3 \vec{b} - 2 \vec{a}$

Hence, If $\vec{a} and \vec{b}$ are the position vectors of A and B respectively, then the position vector of a point C on AB produced such that $\vec{A C} = 3 \vec{A B} is$ $3 \vec{b} - 2 \vec{a} .$

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