Grade 12 Calculus and Vectors
Chapter 9: Equations of Lines in Two- and Three-Dimensions
In this chapter we will be mainly developing vector and parametric equations of lines in two- and three-dimensional space. Just as two points or a point and a slope are required to determine a scalar equation of a straigh line, commonly known as the equation of straight line. We will need two points or one point and a direction vector to determine a parametric or vector equation of a line. Moreover, there is a simple relationship between slope, direction vector and direction numbers.
Let M = (Mx,My) = (2,4) be a direction vector
Then, direction numbers are Mx = 2 and My = 4
And, slope = m = My/Mx= 4/2 = 2
- A vector used to indicate, or point, in the direction of point two from point one. The direction vector may be relocated to the origin of the coordinate system in which point one and point two exist, and because its a vector, by definition it will still hold its importance, or property, as a direction vector.
Direction vectors can be expressed as unit vectors, but in many instances that may not be the case. Moreover, the close relationship between direction vector and slope may have lead textbooks to indicate direction vectors with the letter m. Please see table below for a comparison of a line and a vector.
- Direction Numbers are the x and y components of a direction vector. Assume a direction vector m, where the tail of the vector is at point a(2,3) and head is at point b(4,6). Therefore, the direction vector would be (4-2,6-3) which is (2,3) pointing from point a to point b (note: this direction vector is not a unit vector as indicated above). As such, the Direction Numbers of direction vector m are 2 and 3.
- Parameters are variables used to relate coordinate variables associated with each axis of a coordinate system. When parameteric variables are used, even the independent variables/axes become dependent variable due to the dependency they have on the parameter variable. In the case of a 2-D coordiinate systems, the variable used to indicate a parameter (extra information) would be seen as the third variable (in the system of eqations), yet it would not produce its own axis or show up on any existing axes.
|Time and angles (commonly denoted by letter t and θ, respectively) are the two most common examples of parameters used in many math problems and coordinate systems. As you may have heard, time is commonly considered as the 4th dimension in our space and time jelly/fabric.|
- First of all, parametric relationships are NOT FUNCTIONS, because all such relationships will fail the vertical line test. Such relationships are expressed through parametric equations, or by introducing a new intra-dependent variable, called parameter (some instances may use more than one parameter, as you will see in the next chapter: Equations of Planes). Moreover, in a system of parametric equations only parameters are considered independent variables.
Parametric Equations are:
1. system of equations used to describe a relation that is not a function.
2. system of equations used to describe each coordinates (variables assigned to each axis), of a curve or surface using independent variables, know as parameters, and direction vectors.
3. in "(mathematics) a set of equations that defines the coordinates of the dependent variables (x, y and z) of a curve or surface in terms of one or more independent variables or parameters" -- http://en.wiktionary.org/wiki/parametric_equation -- [21 Feb 11]
- Geometric and Algebraic Vectors
- Dot Products and Cross Products
- Plotting Points and Vectors in two- and three-dimensions
Comparison and contrast of Vectors and Lines
|- Vectors have a definite magnitude||- Lines have infinite magnitude or length|
|- Vectors have a unique direction||- Lines do not have a unique direction, in fact they have two directions (hence the two arrow heads at either end of each line)|
|- Vectors can be transformed into different locations and still be considered as the exact same vector as long as the magnitude and direction are not altered during the transformation||- Lines have a definite location (each line have specific x- and y-intercepts), once a line has been shifted or transformed to new location. This new line will not be considered the same line as the previous|
|- If two vectors are considered to be the same (also referred to as "equal" vectors), both must have the exact same magnitude and direction||- If two lines are to be considered the same (also referred to as"collinear") both lines must have the same slope and y-intercepts. In other words, one line must lie on top of the other line|
|- Two points or a point and a direction vector are required to determine a vector or parametric equation of a straight line||- Two points or a point and a slope are required to determine a scalar equation of a straight line (or just straight line)|
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