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

Concept: Section Formula (Midpoint)

The midpoint’s position vector is the average of the position vectors of the two endpoints.

Step 1 – Write position vectors

$\vec{OP} = 2\hat{i} + 3\hat{j} + 4\hat{k}$,

$\vec{OQ} = 4\hat{i} + 1\hat{j} - 2\hat{k}$.

Step 2 – Apply midpoint formula

Midpoint vector $= \frac{\vec{OP} + \vec{OQ}}{2}$.

Step 3 – Compute …

Why the midpoint formula works

When you have two points in space, the vector from the origin to the midpoint is simply the average of the two position vectors. Think of it this way: if you walk from $P$ to $Q$, the midpoint is exactly halfway along that journey. So you start at $\vec{OP}$, then add half of the vector $\vec{PQ}$ (which is $\vec{OQ} - \vec{OP}$). That gives:

This is the Section Formula for the midpoint — a special case of the more general internal division formula where the ratio is $1:1$.

Step-by-step solution

1. Write the position vectors

   For $P(2, 3, 4)$: $\vec{OP} = 2\hat{i} + 3\hat{j} + 4\hat{k}$

   For $Q(4, 1, -2)$: $\vec{OQ} = 4\hat{i} + 1\hat{j} - 2\hat{k}$

2. Add the vectors component-wise

3. Divide by 2 …

Method: Midpoint of a Segment via Position Vectors

Use this when asked for the midpoint of the segment joining two points — the $1:1$ special case of the section formula.

Steps

Step 1: Write both endpoints as position vectors.

For $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$:

Step 2: Average the two position vectors.

$\vec{OM}=\frac{\vec{OP}+\vec{OQ}}{2}$ …

Common Mistakes

Mistake 1: Mishandling a negative coordinate.

Why it's wrong: with $Q$ having $z=-2$, the $z$-sum is $4+(-2)=2$, not $4+2=6$. Correct approach: add coordinates as signed numbers, so the $\hat k$ term is $\tfrac{2}{2}=1$.

Mistake 2: Adding the position vectors but forgetting to divide by 2.

Why it's wrong: $\vec{OP}+\vec{OQ}=6\hat i+4\hat j+2\hat k$ is twice the midpoint vector, not the midpoint. Correct approach: divide the sum by $2$. …