Chapter 1: Vectors, Lines and Planes
Section 1.2: Vector Arithmetic
Example 1.2.10
Obtain PQ→, the vector from P:2,3,−4 to Q:−5,1,7, then draw it and the position vectors to points P and Q.
Solution
Mathematical Solution
In Figure 1.2.10(a), the red and green arrows represent the position vectors to points P and Q, respectively.
The black arrow, PQ→, the vector from point P to point Q, is the requisite vector.
The gray vector, PQ→#, is the translation of PQ→ back to the origin.
Vector addition gives PQ→′+P=Q, so PQ→# = Q−P.
Translating PQ→# yields PQ→, the vector from point P to point Q.
use plots, Student:-VectorCalculus in module() local P,Q,PQ,PQO,p1,p2; P:=<2,3,-4>; Q:=<-5,1,7>; PQO:=Q-P; PQ:=RootedVector(root=[2,3,-4],Q-P); p1:=PlotVector([P,Q,PQ,PQO],color=[red,green,black,gray],width=.4); p2:=display(p1,scaling=constrained,labels=[x,y,z],orientation=[-30,75,0],axes=frame,tickmarks=[5,4,10],lightmodel=none,glossiness=0); print(p2); end module: end use:
Figure 1.2.10(a) Vectors P, Q, and PQ→
PQ→=Q−P=−517−23−4=−7−211
Maple Solution - Interactive
Enter the data
Enter position vector as per Table 1.1.1.
Context Panel: Assign to a Name≻P
2,3,−4→assign to a nameP
Context Panel: Assign to a Name≻Q
−5,1,7→assign to a nameQ
Obtain PQ→
Write the difference of position vectors Q and P.
Context Panel: Evaluate and Display Inline
Q−P =
Obtaining Figure 1.2.10(a) interactively requires that the three main arrows be drawn separately, their colors changed interactively via the Context Panel, and combined via drag-and-drop.
Type P, Q, and Q−P, as appropriate, and press the Enter key
Context Panel: Plots≻Arrow from origin (for P and Q) Context Panel: Plots≻Arrow from point (for Q−P)≻2,3,−4
Copy/paste each arrow. Select arrow
Context Panel: Color
P
→plot arrow
Q
Q−P
Changing the color of a three-dimensional arrow drawn via the Context Panel is difficult because the arrow has to be selected via the mouse. The arrow has two parts, the shaft and the head. Naive selection will change the color of one but not the other. Make two separate selections and color-changes per arrow.
Maple Solution - Coded
The following sequence of commands will draw the principal vectors in Figure 1.2.10(a). The RootedVector command creates a vector whose "tail" or root is part of the data structure. The PlotVector command makes provision for assigning arbitrary colors to individual vectors.
P≔2,3,−4:Q≔−5,1,7:PQ≔VectorCalculus:-RootedVectorroot=2,3,−4,Q−P:VectorCalculus:-PlotVectorP,Q,PQ,color=red,green,black,width=.2,scaling=constrained,labels=x,y,z
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