$R$ divides the line joining two points $P$ and $Q$ whose position vectors are $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $-\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ respectively in the ratio $2: 1$ externally. $S$ divides $P Q$ internally in the ratio $2: 1$. Then, the…

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$R$ divides the line joining two points $P$ and $Q$ whose position vectors are $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $-\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ respectively in the ratio $2: 1$ externally. $S$ divides $P Q$ internally in the ratio $2: 1$. Then, the position vector of the midpoint of the line joining $R$ and $S$ is

Aicharya · Accepted Answer

We have, $P=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $Q=-\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ $R$ divides externally in the ratio $2: 1$ $	herefore \quad R=\frac{(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}})-2(-\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})}{1-2}$ $R=\frac{\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}+2 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}}{-1}$ $=-3 \hat{\mathbf{i}}+3 \hat{\mathbf{k}}$ $S$ divides internally in the ratio $2: 1$ $	herefore \quad S=\frac{(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}})+2(-\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})}{2+1}$ $S=\frac{\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}-2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}}{3}=\frac{-\hat{\mathbf{i}}+4 \hat{\mathbf{j}}+\hat{\mathbf{k}}}{3}$ Mid-point of $R$ and $S$, is $\frac{-3 \hat{\mathbf{i}}+3 \hat{\mathbf{k}}+\frac{-I+4 \hat{\mathbf{j}}+\hat{\mathbf{k}}}{3}}{2}$ $=\frac{-9 \hat{\mathbf{i}}+9 \hat{\mathbf{k}}-\hat{\mathbf{i}}+4 \hat{\mathbf{j}}+\hat{\mathbf{k}}}{6}$ $=\frac{-10 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+10 \hat{\mathbf{k}}}{6}=\frac{-5}{3} \hat{\mathbf{i}}+\frac{2}{3} \hat{\mathbf{j}}+\frac{5}{3} \hat{\mathbf{k}}$

$R$ divides the line joining two points $P$ and $Q$ whose position vectors are $\hat{\mathbf{i}}+2…

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