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If $P$ divides the line segment joining the points $A$ and $B$ in the ratio $2: 1$ and the position vectors of $A$ and $B$ are $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}$ and $-3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}$ respectively, then the position vector of $p$ is
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Verified Answer
The correct answer is:
$\frac{-5 \hat{i}+8 \hat{j}}{3}$
It is given that the position vectors of points $A$ and $B$ are $\mathbf{O A}=\hat{\dot{\mathbf{i}}}-2 \hat{\mathbf{j}}$ and $\mathbf{O B}=-3 \hat{\dot{\mathbf{i}}}+5 \hat{\mathrm{j}}$.
So, position vector of point $P$. Which divides the line segment joining $A$ and $B$ in the ratio $2: 1$ is
$$
\mathbf{O P}=\frac{1(\hat{\mathbf{i}}-2 \hat{\mathbf{j}})+2(-3 \hat{\mathbf{i}}+5 \hat{\mathrm{j}})}{1+2}=\frac{-5 \hat{\mathrm{i}}+8 \hat{\mathrm{j}}}{3}
$$
So, position vector of point $P$. Which divides the line segment joining $A$ and $B$ in the ratio $2: 1$ is
$$
\mathbf{O P}=\frac{1(\hat{\mathbf{i}}-2 \hat{\mathbf{j}})+2(-3 \hat{\mathbf{i}}+5 \hat{\mathrm{j}})}{1+2}=\frac{-5 \hat{\mathrm{i}}+8 \hat{\mathrm{j}}}{3}
$$
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