Chapter 1: Vectors, Lines and Planes
Section 1.5: Applications of Vector Products
Example 1.5.8
Use the appropriate formula from Table 1.5.1 to calculate the distance of the point P:2,−3,4 to the plane through the three points Q:1,2,−3, R:5,4,7, and S:6,−5,−1.
Solution
Mathematical Solution
Figure 1.5.8(a) displays the points P, Q, R, S, the plane containing Q, R, and S, and the vectors
A=R−Q = 547−12−3 = 4210
B=S−Q = 6−5−1−12−3 = 5−72
C=P−Q = 2−34−12−3 = 1−57
Figure 1.5.8(a) Points P, Q, R, S, and vectors A, B, C
where P, Q, R, and S are respectively the position vectors to points P, Q, R, and S.
The distance from point P to the plane containing points Q, R, and S, is given by ABCA×B. Since
ABC = |42105−721−57| = −402
A×B = |ijk42105−72| = 7442−38
A×B
=742+422+−382
=8684
=22171
the requisite distance is −40222171=2012171 ≐4.31.
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Define the position vectors P, Q, R, and S
Enter P as per Table 1.1.1.
Context Panel: Assign to a Name≻P
2,−3,4→assign to a nameP
Enter Q as per Table 1.1.1.
Context Panel: Assign to a Name≻Q
1,2,−3→assign to a nameQ
Enter R as per Table 1.1.1.
Context Panel: Assign to a Name≻R
5,4,7→assign to a nameR
Enter S as per Table 1.1.1.
Context Panel: Assign to a Name≻S
6,−5,−1→assign to a nameS
By subtraction, obtain the vectors A, B, and C
Context Panel: Assign Name
A=R−Q→assign
B=S−Q→assign
C=P−Q→assign
Apply the appropriate distance formula from Table 1.5.1
Common Symbols palette: Dot- and cross-product operators Use A·B×C for the triple scalar product ABC.
Keyboard vertical strokes for absolute values and norms.
Context Panel: Evaluate and Display Inline
Context Panel: Approximate≻5 (digits)
|A·B×CA×B| = 20121712171→at 5 digits4.3139
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the position vectors P, Q, R, and S.
P,Q,R,S≔2,−3,4,1,2,−3,5,4,7,6,−5,−1:
Define the vectors A, B, C by subtraction.
A,B,C≔R−Q,S−Q,P−Q:
Apply the abs, BoxProduct, Norm, and CrossProduct commands.
absBoxProductA,B,C/NormCrossProductA,B = 20121712171
Solution from first principles
Apply the abs command (for absolute value).
Apply the DotProduct, CrossProduct, and Norm commands. Press the Enter key.
Apply the evalf command to obtain a floating-point (decimal) approximation.
d≔absDotProductA,CrossProductB,CNormCrossProductA,B = 20121712171
evalfd = 4.313860984
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