Chapter 1: Vectors, Lines and Planes
Section 1.3: Dot Product
Suppose the components of the planar vectors A and B are functions of t, and the derivative of such vectors is defined to be the vector of componentwise derivatives. If the prime denotes differentiation with respect to t, show that
The left-hand side of the given identity is expanded as follows.
=ddta1 b1+a2 b2
=a1 b1/+a1/ b1+a2 b2/+a2/ b2
The right-hand side of the given identity is expanded as follows.
Even casual observation confirms that the two expressions agree.
=2 a1a1/+2 a2a2/
With the Student MultivariateCalculus package loaded, the differentiation operator automatically maps over all the components of a vector.
Tools≻Load Package: Student Multivariate Calculus
Write vector A as per Table 1.1.1.
Context Panel: Assign Name
Write vector B as per Table 1.1.1.
Implement the calculation on the left-hand side of the given identity
Calculus palette: Differentiation template
Common Symbols palette: Dot-product operator
Press the Enter key.
ⅆⅆ t A·B
Implement the calculation on the right-hand side of the given identity
ⅆⅆ t A·B+A·ⅆⅆ t B
The expanded forms of the left and right sides of the given identity are indeed identical.
As in Part (a), install the Student MultivariateCalculus package and define the vectors A and B.
ⅆⅆ t A·A
2 A·ⅆⅆ t A
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