Chapter 1: Vectors, Lines and Planes
Section 1.5: Applications of Vector Products
Example 1.5.13
The force F=2 i−3 j+4 k is applied to the head of the position vector r=3 i+2 j−5 k.
Find τ, the torque vector, and τ, its magnitude.
What is the angle between F and r?
Find a unit vector in the direction of the axis of rotation.
Solution
Mathematical Solution
Part (a)
τ=r×F = |ijk32−52−34| = 8−15−(12−(−10))−9−4 = −7−22−13
τ=τ = −72+−222+−132 = 702 = 378
Part (b)
cosθ=F·rF r = −2029 38 = −201102
θ=cos−1−201102=π−cos−1201102 ≐ 2.22 (radians)
Part (c)
The torque vector τ lies along the axis of rotation.
A unit vector in this direction is
ττ = 1378 −7−22−13
Figure 1.5.13(a) shows r in black, F in green, the axis of rotation as a blue line, and a unit vector along this axis in green.
Figure 1.5.13(a) Vectors r, F, and τ/τ
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Enter F as per Table 1.1.1.
Context Panel: Assign to a Name≻F
2,−3,4→assign to a nameF
Enter r as per Table 1.1.1.
Context Panel: Assign to a Name≻r
3,2,−5→assign to a namer
Calculate τ, the torque vector
Common Symbols palette: Cross product operator
τ=r×F→assign
Display τ, the torque vector
Write τ.
Context Panel: Evaluate and Display Inline
τ =
Obtain τ=τ
Keyboard the norm bars.
τ = 378
Write the sequence of two vectors. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Angle
Context Panel: Approximate≻5 (digits)
F,r = →angleπ−arccos105512938→at 5 digits2.2174
Write τ Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Normalize
τ = →normalize
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the vectors F and r.
F,r≔2,−3,4,3,2,−5:
Apply the CrossProduct command.
τ≔CrossProductr,F
Apply the Norm command.
Normτ = 378
Apply the Angle command.
θ≔AngleF,r
π−arccos105512938
Apply the evalf command.
evalfθ,5 = 2.2174
Apply the Normalize command.
Normalizeτ
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