Chapter 1: Vectors, Lines and Planes
Section 1.3: Dot Product
Do the terminal points of the position vectors to points A:2,−3,7, B:1,−1,10, and C:3,−5,4 lie on a straight line?
Guided by Figure 1.3.13(a) in which the position vectors A, B, and C appear in black, red, and green, respectively, obtain vectors from B to A, and A to C.
The vectors A−B and C−A, drawn in blue and gold, respectively, are both i−2 j−3 k, and hence, are collinear.
Thus, the tips of the vectors A, B, and C all lie along a straight line.
Figure 1.3.13(a) Position vectors A, B, and C, along with vectors A−B and C−A
Maple Solution - Interactive
Tools≻Load Package: Student Multivariate Calculus
Enter vector A as per Table 1.1.1.
Context Panel: Assign to a Name≻A
2,−3,7→assign to a nameA
Enter vector B as per Table 1.1.1.
Context Panel: Assign to a Name≻B
1,−1,10→assign to a nameB
Enter vector C as per Table 1.1.1.
Context Panel: Assign to a Name≻C
3,−5,4→assign to a nameC
Determine the angle between the vectors A−B and C−A
Write the sequence of vectors A−B,C−A
Press the Enter key.
Context Panel: Student Multivariate Calculus≻
Lines & Planes≻Angle
Since A−B = C−A, these two vectors are collinear, and hence the tips of vectors A, B, and C must lie on a straight line. Alternatively, the angle between the vectors A−B and C−A is zero, and the same conclusion can be drawn.
Maple Solution - Coded
Load the Student MultivariateCalculus package.
Define the vectors A, B, and C.
Apply the Angle command.
AngleA−B,C−A = 0
From first principles, compute the cosine of the angle between the vectors A−B and C−A.
Apply the DotProduct and Norm commands, as appropriate.
DotProductA−B,C−ANormA−B NormC−A = 1
If the cosine of the angle between the vectors A−B and C−A is 1, then the angle must be zero, as found by application of the Angle command.
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