Chapter 1: Vectors, Lines and Planes
Section 1.1: Cartesian Coordinates and Vectors
Essentials
In Cartesian coordinates, the point P:a,b,c, whose coordinates are a,b,c, is identified with the vector
P=abc
whose components are a,b,c. In this Study Guide, vectors will be denoted in a bold, Roman typeface. (A common alternative notation is P→.) The vector P is represented by an arrow from the origin to point P; this arrow is called the position vector to point P.
The length of the arrow corresponding to the vector P is the magnitude of P, denoted by the symbol P, which is typically called the norm of P. Since there are different kinds of norms, the one used throughout this Study Guide is called the Euclidean norm, and is given by
P = a2+b2+c2
Because each component of the vector is squared, this norm is also called the 2-norm. Note also that some texts will use the notation P to designate the length of the vector. Maple will compute the length of the vector only if the "double bars " used. In Maple, with one of the packages Student MultivariateCalculus, Student LinearAlgebra, Student VectorCalculus, or VectorCalculus loaded, the notation P will default to the Euclidean norm of the vector P. In all other cases the Euclidean norm is obtained with the notation P2 or with the Context Panel option Norm≻Euclidean.
In addition to its magnitude, a vector also has a direction. In the plane, the Cartesian vector given by
V=ab
has direction θ=arctanb,a, where θ∈−π,π is the angle the vector makes with the positive x-direction. (See Figure 1.1.1.)
use plots, Student:-VectorCalculus in module() local p1,p2,p3,p4; p1:=PlotVector(<1,2>,color=black); p2:=plot([[[0,0],[1,0]],[[1,0],[1,2]]],style=line,linestyle=dot,color=red): p3:=textplot({[.5,.1,typeset(a)],[.9,1,typeset(b)],[.2,.1,typeset(theta)]}); p4:=display(p1,p2,p3,scaling=constrained,axes=none); print(p4); end module: end use:
Figure 1.1.1 The direction of a plane vector
The direction of the vector P in Figure 1.1.2 is given by its three direction angles α,β,γ, the respective angles made by the vector and the coordinate axes.
The right triangle with the red leg in Figure 1.1.2 determines α; with the green leg, β; and with the blue leg, γ. The vector itself is the hypotenuse of each of these triangles. (Rotate the figure to see each of these three triangles clearly.)
The cosines of the direction angles are called the direction cosines.
Multiplying the direction cosines by a common factor results in a set of direction numbers.
use plots, Student:-VectorCalculus in module() local p1,p2,p3,p4,p5; p1:=PlotVector(<1,2,3>,color=black); p2:=spacecurve([1,2*t,3*t],t=0..1,numpoints=2,color=red): p3:=spacecurve([t, 2, 3*t],t=0..1,numpoints=2,color=green): p4:=spacecurve([t, 2*t, 3],t=0..1,numpoints=2,color=blue,thickness=2); p5:=display(p1,p2,p3,p4,scaling=constrained,axes=normal,labels=[x,y,z],tickmarks=[0,0,0],orientation=[35,80,0],lightmodel=none); print(p5); end module: end use:
Figure 1.1.2 The direction of the vector P
The end of the position vector at the origin is called the tail, whereas the point of the arrow is called the head or tip. A vector whose tail is fixed to a point is called a bound vector. A Cartesian vector can be transported (slid) parallel to its original direction so that its tail is now at some point other than the origin. Such "movable" vectors are called free vectors.
Table 1.1.1 details two different ways to define a vector in Maple. By default, vectors defined in either of these two ways are treated as position vectors, and if drawn, are drawn as arrows from the origin.
Technique
Details
Interactive
Use the Matrix palette. (See Figure 1.1.3.)
Figure 1.1.3 The Matrix palette
The image on the left in Figure 1.1.3 shows the default state of the Matrix palette. The image in the middle is obtained by changing the number of rows to 3 and the number of columns to 1. This can be done interactively via the Choose button that allows for a dynamic setting of the shape of the item to be inserted by the palette.
Click the "Insert Vector [column]" button to obtain the template shown on the right in Figure 1.1.3. Type the components of the vector, using the Tab key to advance to the next field.
Coded
Write a,b,c, using the inequality symbols for the "wedge brackets".
Pressing the Enter key will echo the vector as a column vector.
Table 1.1.1 Defining vectors in Maple
Examples
Example 1.1.1
Draw the position vector to the Cartesian point 1,2.
Example 1.1.2
Calculate the magnitude of the position vector to the point 1,2.
Example 1.1.3
Draw the position vector to 1,2 translated to the point −2,−1.
Example 1.1.4
Determine the angle that the position vector to 1,2 makes with the x-axis.
Example 1.1.5
Draw the position vector to the Cartesian point 1,2,3.
Example 1.1.6
Calculate the magnitude of the position vector to the point 1,2,3.
Example 1.1.7
Draw the position vector to 1,2,3 translated to the point 2,2,−3.
Example 1.1.8
Determine the angles α,β,γ, that the position vector to 1,2,3 makes with the x-, y-, and z-axes, respectively.
Example 1.1.9
Determine the coordinates of the tip of the position vector to 1,2 if it is translated so its tail is at the point 3,−4.
Example 1.1.10
Determine the coordinates of the tip of the position vector to 1,2,3 if it is translated so its tail is at the point 3,2,1.
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