With i east and j north, speed s on bearing beta gives v=ssinbeta, i+scosbeta, j. A bearing is clockwise from north. The angle is measured from the northward axis, so the north component is adjacent to it. Component signs locate the correct compass quadrant. Draw the direction; label east/north signs; resolve using the angle actually supplied. To recover a bearing, find an acute reference angle from component magnitudes and place it in the correct quadrant. For v=3 i-4 j, speed is 5. The acute angle east of south is tan-1((3)/(4))=36.87, so the bearing is 180 -36.87 =143.13. Exam check: tan-1(y/x) alone does not determine a compass bearing. Check both signs; a westward horizontal vector has bearing 270, not 90.

With \(\mathbf i\) east and \(\mathbf j\) north, speed \(s\) on bearing \(\beta\) gives \(\mathbf v=s\sin\beta\,\mathbf i+s\cos\beta\,\mathbf j\). A bearing is clockwise from north.

The angle is measured from the northward axis, so the north component is adjacent to it. Component signs locate the correct compass quadrant.

Draw the direction; label east/north signs; resolve using the angle actually supplied. To recover a bearing, find an acute reference angle from component magnitudes and place it in the correct quadrant.

For \(\mathbf v=3\mathbf i-4\mathbf j\), speed is \(5\). The acute angle east of south is \(\tan^{-1}\left(\frac{3}{4}\right)=36.87\ldots^\circ\), so the bearing is \(180^\circ-36.87\ldots^\circ=143.13\ldots^\circ\).

Exam check: \(\tan^{-1}\left(\dfrac{y}{x}\right)\) alone does not determine a compass bearing. Check both signs; a westward horizontal vector has bearing \(270^\circ\), not \(90^\circ\).

Components and compass bearings

Components and compass bearings