Compass directions and vectors

Learning Objectives Convert compass directions and bearings into standard mathematical angles. Represent a displacement described by a magnitude and bearing as a vector in component form. Use trigonometry (sine and cosine) to resolve a vector into its horizontal (East-West) and vertical (North-South) components. Add two or more vectors in component form to find a resultant vector. Calculate the magnitude and bearing of a resultant vector from its components. Solve multi-step word problems involving navigation, such as finding the final position of a ship or aircraft. Ever wonder how a pilot navigates a plane or a captain steers a ship without any roads? ✈️ They use the power of vectors and compass directions to find their way! This tutorial connects the mathematical concept...

TermDefinitionExample VectorA quantity that has both magnitude (size or length) and direction.A displacement of 10 km on a bearing of 090° (due East). MagnitudeThe size, length, or amount of a vector. It is always a non-negative value.For a velocity vector of 50 km/h North, the magnitude is 50 km/h. Bearing (True Bearing)The direction of a vector, measured as an angle in degrees clockwise from the North direction. It is always written using three figures.North is 000°, East is 090°, South is 180°, and West is 270°. A direction of North-East is a bearing of 045°. Component FormA way of writing a vector by its horizontal (x) and vertical (y) components. The x-component represents East-West movement and the y-component represents North-South movement.A vector representing a movement of 4 uni...

Components from Magnitude and Bearing For a vector `\vec{v}` with magnitude `|\vec{v}|` and bearing `β`: Horizontal component: `v_x = |\vec{v}| \sin(β)` Vertical component: `v_y = |\vec{v}| \cos(β)` Use these formulas to convert a vector from magnitude-bearing form into its x (East/West) and y (North/South) components. A positive `v_x` is East, negative is West. A positive `v_y` is North, negative is South. Magnitude and Bearing from Components For a vector `\vec{v} = (v_x, v_y)`: Magnitude: `|\vec{v}| = \sqrt{v_x^2 + v_y^2}` Reference Angle: `α = \tan^{-1}(|v_x / v_y|)` Use `α` and the signs of `v_x` and `v_y` to find the bearing `β`. After finding a resultant vector in component form, use the Pythagorean theorem to find its total length (magnitude). Use the inverse tangent...