Position Vector of Midpoint of Line
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Theorem
Let $\mathbf a$ and $\mathbf b$ be the position vectors of points $A$ and $B$.
The position vector $\mathbf r$ of the midpoint of the line segment $AB$ is given by:
- $\mathbf r = \dfrac {\mathbf a + \mathbf b} 2$
Proof
From Point dividing Line Segment between Two Points in Given Ratio:
- the position vector $\mathbf r$ of a point $R$ on $AB$ which divides $AB$ in the ratio $m : n$ is given by:
- $\mathbf r = \dfrac {n \mathbf a + m \mathbf b} {m + n}$
In this case the ratio $m : n$ is $1 : 1$.
Hence when $\mathbf r$ is the position vector of the midpoint of $AB$:
\(\ds \mathbf r\) | \(=\) | \(\ds \dfrac {1 \times \mathbf a + 1 \times \mathbf b} {1 + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\mathbf a + \mathbf b} 2\) |
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bisect
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bisect