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Question

(A) The position vector of a point $R$ dividing the line segment joining two points $P$ and $Q$ with position vectors $\vec{a}$ and $\vec{b}$ in the r

Vedclass · Accepted Answer

$-\frac{1}{3} \hat{i}+\frac{4}{3} \hat{j}+\frac{1}{3} \hat{k}$

Vedclass · Answer

(A) The position vector of a point $R$ dividing the line segment joining two points $P$ and $Q$ with position vectors $\vec{a}$ and $\vec{b}$ in the ratio $m:n$ internally is given by the formula: $\vec{r} = \frac{m\vec{b} + n\vec{a}}{m+n}$.Here,$\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$,$\vec{b} = -\hat{i} + \hat{j} + \hat{k}$,$m = 2$,and $n = 1$.Substituting these values into the formula:$\vec{r} = \frac{2(-\hat{i} + \hat{j} + \hat{k}) + 1(\hat{i} + 2\hat{j} - \hat{k})}{2+1}$$\vec{r} = \frac{(-2\hat{i} + 2\hat{j} + 2\hat{k}) + (\hat{i} + 2\hat{j} - \hat{k})}{3}$$\vec{r} = \frac{(-2+1)\hat{i} + (2+2)\hat{j} + (2-1)\hat{k}}{3}$$\vec{r} = \frac{-\hat{i} + 4\hat{j} + \hat{k}}{3} = -\frac{1}{3}\hat{i} + \frac{4}{3}\hat{j} + \frac{1}{3}\hat{k}$.

Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $\hat{i}+2 \hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively,in the ratio $2: 1$ internally.

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