University:
Arizona State University
Course:
PHYS 102 | Introductory Physics I
Academic year:

2024
Views:

493

Pages:

6
Author:

Mohammed N.

Vector Scalar A scalar has only magnitude. It does not have direction. Some examples are mass, distance, speed, time, work, energy, temperature, volume, and pressure, etc. m = 50kg s = 10km v = 20m / s t = 10 s W = 15J E = 100J T = 30K Vector in 2D A vector has both magnitude and direction. It has two kinds of representations: polar representation and component representation. Each has its convenience for solving different problems. The following equations are both for 2D vector representations. Polar: vec A = (A, theta) Component: vec A =A x hat i +A y hat j Vector in 2D Polar Coordinates For a 2D vector in polar representation, we can calculate its polar coordinates from the vector's Cartesian coordinates. vec A = (A, theta) A = sqrt(A_{x} ^ 2 + A_{y} ^ 2); tan theta = A_{y}/A_{x} 0 < theta < pi/2 * (A_{x} > 0, A_{y} > 0) <0 < π(Α < 0, Α₁ > 0). π tau < theta < (3pi)/2 * (A_{x} < 0, A_{y} < 0) (3pi)/2 < theta <2 pi(A x >0 , Α <0 theta = pi/2 * (A_{x} = 0, A_{y} > 0) theta = (3pi)/2 * (A_{x} = 0, A_{y} < 0) . Vector in 2D Cartesian Coordinates A vector can be decomposed into components along independent directions. This is called component representation of vectors. The following equations also give us the conversion from polar representation to component representation in 2D. vec A =A x hat i +A y hat j A_{x} = A * cos theta A_{y} = A * sin theta Vector in 3D Spherical Coordinates For a 3D vector in spherical coordinates representation, we can calculate its spherical coordinates from the vector's Cartesian coordinates. The spherical coordinates include magnitude of the vector and two angles. vec A = (A, varphi, theta) A = sqrt(A_{x} ^ 2 + A_{y} ^ 2 + A_{z} ^ 2) tan varphi = (sqrt(A_{x} ^ 2 + A_{y} ^ 2))/A_{z} tan theta = A_{y}/A_{x} Vector in 3D Cartesian Coordinates For a 3D vector in component representation, we can calculate its Cartesian coordinates from the vector's spherical coordinates. vec A =A x hat i +A y hat j +A z hat k A_{x} = A * sin varphi * cos theta A_{y} = A * sin varphi * sin theta A_{z} = A * cos varphi Negative Vector A negative vector keeps the same magnitude as the original vector but points in the opposite direction. Representations of a negative vector in 2D are given below in both polar and component forms. - vec A = (A, pi + theta) - vec A =-A x hat i -A y hat j