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  • If a and b are the position vectors of A and B, respectively, find the position vector of a point C in BA produced such that BC = 1.5 BA.

If \vec{a}  and  \vec{b} are the position vectors of A and B, respectively, find the position vector of a point C in BA produced such that BC = 1.5 BA.

Answers (1)

We have been given that,

Position vector of A =\vec{a}

\begin{aligned} &\Rightarrow \overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{a}}\\ &\text { Position vector of } B=\overrightarrow{\mathrm{b}} \end{aligned}

\Rightarrow \overrightarrow{\mathrm{OB}}=\overrightarrow{\mathrm{b}} \\ \text{Here, O is the origin.}\\ \text{We need to find the position vector of } \mathrm{C},\\ \text{that is}, \overrightarrow{\mathrm{OC}}\\ \text{Also, we have}

\\ \overrightarrow{\mathrm{BC}}=1.5 \overrightarrow{\mathrm{BA}}_{\ldots .(\mathrm{i})} \\ Here, \overrightarrow{\mathrm{BC}}= \text{Position vector of} \mathrm{C} - \text{Position vector of} \mathrm{B}\\ \Rightarrow \overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{OC}}-\overrightarrow{\mathrm{OB}}...(2)

\\ And, \overrightarrow{\mathrm{BA}}= \text{Position vector of A}-\text{Position vector of} \mathrm{B}\\ \Rightarrow \overrightarrow{\mathrm{BA}}=\overrightarrow{\mathrm{OA}}-\overrightarrow{\mathrm{OB}}_{\ldots(\mathrm{iii})}

\\ \overrightarrow{\mathrm{BC}}=1.5 \overrightarrow{\mathrm{BA}} \\ \Rightarrow \overrightarrow{\mathrm{OC}}-\overrightarrow{\mathrm{OB}}=1.5(\overrightarrow{\mathrm{OA}}-\overrightarrow{\mathrm{OB}}) \\ \Rightarrow \overrightarrow{\mathrm{OC}}-\overrightarrow{\mathrm{OB}}=1.5 \overrightarrow{\mathrm{OA}}-1.5 \overrightarrow{\mathrm{OB}} \\ \Rightarrow \overrightarrow{\mathrm{OC}}=1.5 \overrightarrow{\mathrm{OA}}-1.5 \overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OB}}

\\ \begin{aligned} &\Rightarrow \overrightarrow{\mathrm{OC}}=1.5 \overrightarrow{\mathrm{OA}}-0.5 \overrightarrow{\mathrm{OB}}\\ &\Rightarrow \overrightarrow{\mathrm{OC}}=1.5 \overrightarrow{\mathrm{a}}-0.5 \overrightarrow{\mathrm{b}}\\ &[\because \text { it is given that } \overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{a}} \text { and } \overrightarrow{\mathrm{OB}}=\overrightarrow{\mathrm{b}}]\\ &\Rightarrow \overrightarrow{\mathrm{OC}}=\frac{15 \overrightarrow{\mathrm{a}}}{10}-\frac{5 \overrightarrow{\mathrm{b}}}{10} \end{aligned}

\\ \Rightarrow \overrightarrow{\mathrm{OC}}=\frac{3 \overrightarrow{\mathrm{a}}}{2}-\frac{\overrightarrow{\mathrm{b}}}{2} \\ \Rightarrow \overrightarrow{\mathrm{OC}}=\frac{3 \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}}{2}

Thus, position vector of point C is  =\frac{3 \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}}{2}

Posted by

infoexpert22

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