Question

# If $\stackrel{\to }{a}and\stackrel{\to }{b}$ are the position vectors of A and B respectively, then the position vector of a point C on AB produced such that $\stackrel{\to }{AC}=3\stackrel{\to }{AB}\mathrm{is}______________________.$

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Solution

## Given: $\stackrel{\to }{AC}=3\stackrel{\to }{AB}$ Let $\stackrel{\to }{c}$ is the position vectors of C. $\stackrel{\to }{AC}=3\stackrel{\to }{AB}\phantom{\rule{0ex}{0ex}}⇒\stackrel{\to }{c}-\stackrel{\to }{a}=3\left(\stackrel{\to }{b}-\stackrel{\to }{a}\right)\phantom{\rule{0ex}{0ex}}⇒\stackrel{\to }{c}-\stackrel{\to }{a}=3\stackrel{\to }{b}-3\stackrel{\to }{a}\phantom{\rule{0ex}{0ex}}⇒\stackrel{\to }{c}=3\stackrel{\to }{b}-3\stackrel{\to }{a}+\stackrel{\to }{a}\phantom{\rule{0ex}{0ex}}⇒\stackrel{\to }{c}=3\stackrel{\to }{b}-2\stackrel{\to }{a}$ Hence, If $\stackrel{\to }{a}\mathrm{and}\stackrel{\to }{b}$ are the position vectors of A and B respectively, then the position vector of a point C on AB produced such that $\stackrel{\to }{AC}=3\stackrel{\to }{AB}\mathrm{is}$ $\overline{)3\stackrel{\to }{b}-2\stackrel{\to }{a}}.$

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