Question

The position vector of the midpoint of the line joining the points $P(2,3,4)$ and $Q(4,1,-2)$ is

• A. $3\hat{i} + 2\hat{j} + \hat{k}$
• B. $3\hat{i} + 2\hat{j} - \hat{k}$
• C. $\hat{i} - \hat{j} - 3\hat{k}$
• D. $-\hat{i} + \hat{j} + 3\hat{k}$

Hint:

The midpoint of two points in 3D is obtained by averaging their corresponding coordinates.

Answer 1

The Correct Option is A

Step 1: Recall the midpoint formula in three dimensions.
If two points are \[ P(x_1,y_1,z_1) \] and \[ Q(x_2,y_2,z_2) \] then the midpoint is given by \[ \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right) \]
Step 2: Substitute the coordinates.
For \[ P(2,3,4), \quad Q(4,1,-2) \] Compute each coordinate of the midpoint.
\[ x = \frac{2+4}{2} = 3 \] \[ y = \frac{3+1}{2} = 2 \] \[ z = \frac{4+(-2)}{2} = 1 \] Thus the midpoint is \[ (3,2,1) \]
Step 3: Write the position vector.
The position vector corresponding to the point $(3,2,1)$ is \[ 3\hat{i} + 2\hat{j} + \hat{k} \]
Step 4: Conclusion.
Hence the position vector of the midpoint of the given line segment is \[ 3\hat{i} + 2\hat{j} + \hat{k} \] Final Answer: $\boxed{3\hat{i} + 2\hat{j} + \hat{k}}$