Finding the Position Vector of a Midpoint

In vector algebra, the position vector of a point gives its location relative to a fixed origin. If we have two points, say A and B, their position vectors are typically denoted by \(\mathop a\limits^ \to\) and \(\mathop b\limits^ \to\), respectively. These vectors originate from the origin and point to the respective points A and B.

Understanding the Midpoint Formula in Vector Algebra

We want to find the position vector of the midpoint of the line segment AB. Let M be the midpoint of AB, and let its position vector be \(\mathop m\limits^ \to\). The midpoint M is exactly halfway between points A and B.

Geometrically, the vector from the origin to M, \(\mathop m\limits^ \to\), can be found using the property that M divides the segment AB in the ratio 1:1. Using the section formula for vectors, if a point divides the line segment joining points with position vectors \(\mathop p\limits^ \to\) and \(\mathop q\limits^ \to\) in the ratio \(m:n\), the position vector of the dividing point is given by:

\(\frac{{n\mathop p\limits^ \to + m\mathop q\limits^ \to }}{{m + n}}\)

For a midpoint, the ratio \(m:n\) is \(1:1\). Substituting \(m=1\) and \(n=1\) into the section formula, we get the position vector of the midpoint M:

\(\mathop m\limits^ \to = \frac{{1 \cdot \mathop a\limits^ \to + 1 \cdot \mathop b\limits^ \to }}{{1 + 1}}\)

\(\mathop m\limits^ \to = \frac{{\mathop a\limits^ \to + \mathop b\limits^ \to }}{2}\)

\(\mathop m\limits^ \to = \frac{1}{2}\left( {\mathop a\limits^ \to + \mathop b\limits^ \to } \right)\)

This formula is the standard midpoint formula in vector algebra. It tells us that the position vector of the midpoint is the average of the position vectors of the two endpoints. This is a fundamental concept often used in vector calculus.

Applying the Formula to Find the Midpoint Position Vector

Given that the position vector of A is \(\mathop a\limits^ \to\) and the position vector of B is \(\mathop b\limits^ \to\), the position vector of the midpoint of AB is directly given by the formula we derived:

Position vector of midpoint \( = \frac{1}{2}\left( {\text{Position vector of A} + \text{Position vector of B}} \right)\)

Position vector of midpoint \( = \frac{1}{2}\left( {\mathop a\limits^ \to + \mathop b\limits^ \to } \right)\)

This result matches one of the provided options.

Evaluating the Options

Let's compare our result with the given options:

• Option 1: \(\mathop a\limits^ \to + \mathop b\limits^ \to\). This represents the diagonal of a parallelogram formed by \(\mathop a\limits^ \to\) and \(\mathop b\limits^ \to\), not the midpoint.
• Option 2: \(\mathop a\limits^ \to - \mathop b\limits^ \to\). This represents the vector from B to A (\(\vec{BA}\)).
• Option 3: \(\frac{1}{2}\left( {\mathop a\limits^ \to + \mathop b\limits^ \to } \right)\). This matches our derived midpoint formula.
• Option 4: \(\frac{1}{2}\left( {\mathop a\limits^ \to - \mathop b\limits^ \to } \right)\). This would be half the vector from B to A.

Therefore, the correct option for the position vector of the mid-point of AB is \(\frac{1}{2}\left( {\mathop a\limits^ \to + \mathop b\limits^ \to } \right)\). This formula is crucial for solving various problems in vector algebra and related fields like vector calculus.