![]()
![]()
Coaster Dynamics
Physics Primer
Chapter 3
Vectors
Vector and Scalar Quantities
In science, some physical quantities have a "natural direction" associated with them. For example, a force or velocity. On the other hand, other quantities have no direction associated with them -- for example, mass or length. For analyzing quantities with a direction associated with them, vectors are extremely useful.
Vector quantities have both a size, called its magnitude, and a direction. Quantities that only have a size associated with them are called scalars.
We will not attempt to develop the theory of vectors, leaving that to a good textbook on vector mathematics. Here in the
Physics Primer, we will simply summarize the essential rules of vector math, as they apply to the analysis of 2D and 3D motion.Vector Diagrams
Vectors can be pictorially represented with vector diagrams. Figures 3-A and 3-B show the representation of a vector in 2 and 3 dimensions.
Figure 3-A. Vectors in 2 Dimensions.
Figure 3-B. Vectors in 3 Dimensions.
In a vector diagram, the magnitude of the vector is indicated by its length. For instance, if vector B has twice the magnitude of vector A, then its length is double that of A. The direction of the vector is indicated by an arrow.
Resolution of Vectors in 2 Dimensions
As illustrated in the 2D vector diagram in Figure 3-A, a 2D vector can be considered the sum of 2 other vectors, which have their directions parallel to the x and y coordinate directions (as defined by the coordinate system). The representation of a vector this way is called resolving the vector into its components.
In this document, the magnitude of a vector will be printed as a simple text character, with no italics or underlinings. A vector is indicated by a bold character. The direction vectors will be indicated with underlined bold characters. The magnitude of the direction component is indicated with a non-bold character and a subscripted direction character. For example:
(4-1) A = xAx + yAy.
From the 2D vector diagram in Figure 3-A, and using basic geometric analysis, it can be shown that the magnitude of a 2D vector is equal to:
(4-2) A = sqrt( Ax² + Ay² ).
The addition of vectors can be performed by summing the individual components:
(4-3) A + B = x( Ax + Bx ) + y( Ay + By ).
Furthermore, when adding vectors, the commutative and associative laws are valid:
(4-4) A + B = B + A (commutative law),
(4-5) A + (B + C) = (A + B) + C (associative law).
Resolution of Vectors in 3 Dimensions
The analysis of 3D vectors can be performed by the straightforward extension of vectors into the third dimension, z.
For the 3D vector
(4-6) A = xAx + yAy + zAz
its magnitude is equal to
(4-7) A = sqrt( Ax² + Ay² + Az² ).
The addition of vectors is still performed by adding the direction components:
(4-8) A + B = x( Ax + Bx ) + y( Ay + By ) + z( Az + Bz )
and the commutative and associative laws remain valid in 3 dimensions.
Furthermore, although not used in the
Physics Primer (since they are not needed for analysis of non-rotational dynamics), it can be shown that the Vector Dot Product and Vector Cross Product are valid if used with a coordinate system that obeys the right-hand rule (which is true for the Global Coordinate System chosen for Coaster Dynamics ). These functions are useful for calculating other, more complicated, physics quantities such as torque or angular momentum.
Copyright © 2001, Cyclone Software, Pleasanton, CA, USA. All rights reserved.