Question

Find the position vector of the midpoint of the vector joining the points $P(2,3,4)$ and $Q(4,1,-2)$.

• (A) $3\hat{i}+2\hat{j}+\hat{k}$
• (B) $-3\hat{i}-2\hat{j}-\hat{k}$
• (C) $-5\hat{i}-3\hat{j}-8\hat{k}$
• (D) $5\hat{i}-3\hat{j}+8\hat{k}$

Answer 1

The Correct Option is A

Explanation:
Given:The position vector of point $P=2\hat{i}+3\hat{j}+4\hat{k}$Position Vector of point $Q=4\hat{i}+\hat{j}-2\hat{k}$As we know,The position vector of $A(x_1,y_1,z_1)$ is given by$\overrightarrow{OA}=x_1\hat{i}+y_1\hat{j}+z_1\hat{k}$ Compute the position vector of the midpoint of $P$ and $Q$ using the formula:If $C(x,y,z)$ is the midpoint of segment $\overrightarrow{AB}$, then we have the position vector of $C=\overrightarrow{OC}=\frac{\overrightarrow{OA}+\overrightarrow{OB}}{2}$$=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2})$The position vector of $R$ which divides $PQ$ in half is given by: $\overrightarrow{r}=\frac{2\hat{i}+3\hat{j}+4\hat{k}+4\hat{i}+\hat{j}-2\hat{k}}{2}$$\overrightarrow{r}=\frac{2\hat{i}+3\hat{j}+4\hat{k}+4\hat{i}+\hat{j}-2\hat{k}}{2}$$\overrightarrow{r}=\frac{6\hat{i}+4\hat{j}+2\hat{k}}{2}$$=3\hat{i}+2\hat{j}+\hat{k}$Hence, the correct option is (A).