Chapter 1: Vectors, Lines and Planes
Section 1.3: Dot Product
If A is a unit vector whose components are functions of t, show that A and A′ are necessarily orthogonal.
Verify this for A=cost i+sint j.
Verify this for A, the normalization of V=t i+t2 j.
Since A is a unit vector, A·A=1. Using the result in Example 1.3.8(b), differentiate both sides to get
By the orthogonality property of the dot product, the vectors A and A′ are necessarily orthogonal.
The following calculations are totally self-sufficient.
Within the Student MultivariateCalculus package, differentiation automatically maps onto the components of vectors.
Tools≻Load Package: Student Multivariate Calculus
Obtain A, the normalized version of the given vector V
Enter the vector V as per Table 1.1.1.
Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Normalize
Context Panel: Simplify≻Assuming Positive
Context Panel: Assign to a Name≻A
t,t2 = tt2→normalizett4+t2t2t4+t2→assuming positive1t2+1tt2+1→assign to a nameA
Obtain and simplify A′
Calculus palette: Differentiation operator
Context Panel: Simplify≻Simplify
Context Panel: Assign to a Name≻dA
ⅆⅆ t A = −tt2+13/21t2+1−t2t2+13/2= simplify −tt2+13/21t2+13/2→assign to a namedA
Verify that A·A′=0
Common Symbols palette: Dot product operator
A·dA = 0
The value of t in the graph in Figure 1.3.9(a) is controlled by the slider to its right. As t varies, the tip of vector A (in red) traces out the blue curve. The derivative A′ (in green) is drawn at the tip of A so that the orthogonality of A and A′ is more readily observed.
Figure 1.3.9(a) The orthogonality of the vectors A (red) and A′ (green)
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