Calculate Closing Error (e)

The closing error $e$ is the magnitude of the resultant error, calculated using the square root of the sum of squares of total latitude and total departure.

$e = \sqrt{(-100)^2 + (50)^2} = \sqrt{10000 + 2500} = \sqrt{12500} \approx 111.80\text{ m}$

Calculate Angle of Closure (θ)

The reduced bearing angle $\theta$ is given by the tangent of the absolute ratio of total departure to total latitude.

$\tan \theta = \left| \frac{\sum D}{\sum L} \right| = \left| \frac{50}{-100} \right| = 0.5 \implies \theta = \tan^{-1}(0.5) = 26.565°\approx 26.56°$

Determine Quadrant and Reduced Bearing

Since total latitude $\sum L$ is negative (South) and total departure $\sum D$ is positive (East), the closing error vector lies in the South-East quadrant (4th quadrant). Therefore, the direction is expressed as $\text{S}26.56°\text{E}$.

$\text{Bearing of closure} = \text{S}26.56°\text{E}$