Chapter 1: Vectors, Lines and Planes
Section 1.5: Applications of Vector Products
Example 1.5.6
Use the appropriate formula from Table 1.5.1 to calculate the distance of the point P:1,2,3 from the line through points Q:5,−3,7 and R:4,1,−6.
Solution
Mathematical Solution
The three points P, Q, and R necessarily lie in a plane. Using the colors green, black, and red, respectively, Figure 1.5.6(a) represents these three points in that plane.
According to Table 1.5.1, the distance from P to the line through Q and R, is given by
∥A×B∥A
where
Figure 1.5.6(a) Points P, Q, and R
A=R−Q = 41−6−5−37 = −14−13 and B=P−Q = 123−5−37 = −45−4
and P, Q, and R, are position vectors to points P, Q, and R, respectively. Vectors A and B appear in Figure 1.5.6(a) as the gray and gold vectors, respectively.
Since A×B = |ijk−14−13−45−4| = −16+65−(4−52)−5+16 = 494811, A×B=4826, and A=186, the required distance is 4826/186 ≐ 5.09.
Mathematical Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Define the position vectors P, Q, and R
Enter P as per Table 1.1.1.
Context Panel: Assign to a Name≻P
1,2,3→assign to a nameP
Enter Q as per Table 1.1.1.
Context Panel: Assign to a Name≻Q
5,−3,7→assign to a nameQ
Enter R as per Table 1.1.1.
Context Panel: Assign to a Name≻R
4,1,−6→assign to a nameR
By subtraction, obtain the vectors A and B
Context Panel: Assign Name
A=R−Q→assign
B=P−Q→assign
Apply the appropriate distance formula from Table 1.5.1
Keyboard the norm bars.
Common Symbols palette: Cross-product operator
Context Panel: Evaluate and Display Inline
Context Panel: Approximate≻5 (digits)
A×B∥A∥ = 11864826186→at 5 digits5.0936
Mathematical Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the position vectors P, Q, and R.
P,Q,R≔1,2,3,5,−3,7,4,1,−6:
Obtain the vectors A and B.
A,B≔R−Q,P−Q:
Apply the Norm and CrossProduct commands to compute the requisite distance.
Apply the evalf command to obtain a floating-point (decimal) approximation.
d≔NormCrossProductA,BNormA
11864826186
evalfd
5.093744847
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