R divides the line joining two points P and Q | vedclass

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Question

(B) Given position vectors are $\vec{p} = \hat{i}+2 \hat{j}-\hat{k}$ and $\vec{q} = -\hat{i}+\hat{j}+\hat{k}$.$R$ divides $PQ$ externally in the ratio

Vedclass · Accepted Answer

$\frac{-5}{3} \hat{i}+\frac{2}{3} \hat{j}+\frac{5}{3} \hat{k}$

Vedclass · Answer

(B) Given position vectors are $\vec{p} = \hat{i}+2 \hat{j}-\hat{k}$ and $\vec{q} = -\hat{i}+\hat{j}+\hat{k}$.$R$ divides $PQ$ externally in the ratio $2:1$. The formula for external division is $\vec{r} = \frac{m\vec{q} - n\vec{p}}{m-n}$.$\vec{r} = \frac{2(-\hat{i}+\hat{j}+\hat{k}) - 1(\hat{i}+2 \hat{j}-\hat{k})}{2-1} = \frac{-2\hat{i}+2\hat{j}+2\hat{k} - \hat{i}-2\hat{j}+\hat{k}}{1} = -3\hat{i}+3\hat{k}$.$S$ divides $PQ$ internally in the ratio $2:1$. The formula for internal division is $\vec{s} = \frac{m\vec{q} + n\vec{p}}{m+n}$.$\vec{s} = \frac{2(-\hat{i}+\hat{j}+\hat{k}) + 1(\hat{i}+2 \hat{j}-\hat{k})}{2+1} = \frac{-2\hat{i}+2\hat{j}+2\hat{k} + \hat{i}+2\hat{j}-\hat{k}}{3} = \frac{-\hat{i}+4\hat{j}+\hat{k}}{3}$.The midpoint of $RS$ is $\frac{\vec{r} + \vec{s}}{2}$.Midpoint $= \frac{(-3\hat{i}+3\hat{k}) + (-\frac{1}{3}\hat{i} + \frac{4}{3}\hat{j} + \frac{1}{3}\hat{k})}{2} = \frac{-\frac{10}{3}\hat{i} + \frac{4}{3}\hat{j} + \frac{10}{3}\hat{k}}{2} = -\frac{5}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{5}{3}\hat{k}$.

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