2D Coordinate Systems & Vectors PDF

# 2D Coordinate Systems & Vectors A coordinate system gives us a frame of reference to describe a system that we would like to analyze. In statics, we normally use orthogonal coordinate systems, where orthogonal means "perpendicular". In an orthogonal coordinate system, the coordinate directions a...

# 2D Coordinate Systems & Vectors A coordinate system gives us a frame of reference to describe a system that we would like to analyze. In statics, we normally use orthogonal coordinate systems, where orthogonal means "perpendicular". In an orthogonal coordinate system, the coordinate directions are perpendicular to each other and thereby independent. The intersection of the coordinate axes is called the origin, and measurements are made from there. Both points and vectors are described with a set of numbers called the coordinates. There are two-dimensional coordinate orthogonal systems: rectangular and polar coordinates. ## Rectangular Coordinates The most important coordinate system is the Cartesian system. It has two coordinate axes that are straight lines rotated 90° apart named x and y. In most cases, the x-axis is horizontal and points to the right, and the y-axis points vertically upward. Points are specified as an ordered pair of coordinate values separated by a comma and enclosed in parentheses, P = (x, y). Similarly, forces and other vectors will be specfied with an ordered pair of scalar components enclosed by angle brackets, F = <Fx, Fy>. An example of rectangular coordinates with coordinates P(2, 2) is shown below: | | | | |---|---|---| | | | | | | **P=(2,2)** | | | 4 | | | | 2 | y | | | | x | | | 2 | | 4| | | **Rectangular coordinates** | | ## Polar Coordinates The polar coordinate system is an alternate orthogonal system in which a point is specified by giving its distance from the origin *r* and θ, an angle measured counter-clockwise from a reference direction, usually the x-axis (positive direction). Points in polar coordinates will be specified as an ordered pair of values separated by a semicolon and enclosed in parentheses, P = (r; θ). ## Coordinate Transformation Rectangular coordinate system can be transformed to polar coordinate system and vice versa; this can be done by converting coordinate points (x, y) into (r, θ) or (r, θ) into (x, y). In order to perform coordinate transformation, the following formulas are used. ### Rectangular to Polar for Points (Given x and y) * r = √(x² + y²) * θ = tan⁻¹(y/x) * P = (r; θ) ### Polar To Rectangular for Points (Given r and θ) * x = r cos θ * y = r sin θ * P = (x, y) ### Rectangular to Polar for Forces (Given Fx and Fy) * F = √(Fx² + Fy²) * θ = tan⁻¹(Fy/Fx) * F = (F, θ) ### Polar To Rectangular for Forces (Given Magnitude & Direction) * Fx = F cos θ * Fy = F sin θ * F = <Fx, Fy> ## Examples ### Ex 2.1: Convert the rectangular coordinates (3, 4) into polar coordinates. * Given: x = 3, y = 4 * r = √(x² + y²) = √(3² + 4²) = √25 = 5 * θ = tan⁻¹(y/x) = tan⁻¹(4/3) = 53.13° * P = (r; θ) = (5; 53.13°) ### Ex 2.2: Given polar coordinates (r, θ) = (6, 30°) find the rectangular coordinates. * Given: r = 6, θ = 30° * x = r cos θ = 6 cos 30° = 5.19 * y = r sin θ = 6 sin 30° = 3 * P = (x, y) = (5.19, 3) ### Ex 2.3: Given the rectangular force components (Fx, Fy) = (8, 6) N. Find the polar coordinates of that force. * Fx = 8N, Fy = 6N * F = √(Fx² + Fy²) = √(8² + 6²) = √100 = 10 N * θ = tan⁻¹(Fy/Fx) = tan⁻¹(6/8) = 36.87° * F = (F, θ) = (10, 36.87°) ### Ex 2.4: Given force in polar coordinates (F, θ) = (15, 45°) find its rectangular components Fx and Fy. * F = 15N, θ = 45° * Fx = F cos θ = 15N cos 45° = 10.6N * Fy = F sin θ = 15 N sin 45° = 10.6N * F = <Fx, Fy> = <10.6N, 10.6N>