Chapter 2: Space Curves
Section 2.3: Tangent Vectors
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Essentials
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Table 2.3.1 lists the essential tangent-vector facts for a curve described by the position vector .
Item
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Fact
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1
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The derivative is taken componentwise.
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2
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The vector is tangent to the curve defined by .
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3
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If is arc length , is the unit tangent vector T.
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If T is the unit tangent vector, , that is, T is orthogonal to its derivative.
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The length of is given by = =
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If the parameter is the time , then is the velocity vector V; and , the speed.
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Table 2.3.1 Essential tangent-vector facts for a curve defined by the position vector .
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The generic parameter along a curve defined parametrically by the position vector is taken as . The derivative of R with respect to its parameter is always a tangent vector. This can be inferred from the definition of the derivative of R.
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=
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This definition also suggests why the derivative is taken componentwise. The difference is a vector whose components are each divided by . As , each component of the fraction becomes the derivative of that component.
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Figure 2.3.1 contains an animation that illustrates why this derivative results in a vector tangent to the curve described by .
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The blue vector is and can be considered a "secant vector" that shrinks in length as .
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The green vector is , and has the same direction as the secant vector, but does not shrink in length. In the limit as , this green vector becomes tangent to the curve R.
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>
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use plots in
module()
local p1,p2,p3,p4,f1,f2,f3,f4,k;
p1 := plot([2+4*cos(t),2+2*sin(t),t=0..1.8],color=black):
p2 := arrow([0,0],[2,4],color=black):
f1 := h -> arrow([0,0],[2+4*cos(Pi/2-h),2+2*sin(Pi/2-h)],color=red):
f2 := h -> arrow([2,4],[4*sin(h),2*cos(h)-2],color=blue):
f3 := h-> arrow([2,4], [4*sin(h)/h,(2*cos(h)-2)/h], color=green):
f4 := h -> display([f1(h),f2(h),f3(h)]):
p3 := display([seq(f4(1.5-k/10),k=0..14),seq(f4(.1-k/100),k=1..9)], insequence=true):
p4 := display([p1,p2,p3],scaling=constrained);
print(p4);
end module:
end use:
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Figure 2.3.1 Animation: is a tangent vector
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Examples
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Example 2.3.1
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If is the position-vector representation of the parametric curve , , , show that , denoted by ,
is the vector . Thus, the differentiation operator is applied to R by applying it to each component of R.
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Example 2.3.2
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If is the position-vector representation of , the parametric curve , , ,
a)
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Obtain and the unit tangent vector .
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b)
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Graph R and the vectors along .
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c)
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Graph R and the vectors along .
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d)
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Show that , thus verifying that a unit vector is necessarily orthogonal to its derivative.
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e)
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To the graph in Part (c), add the vectors .
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Example 2.3.3
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Let be the position-vector representation of the parametric curve , , and let be the reparametrization obtained in Example 2.2.6. (Recall that is the arc length along the curve.)
a)
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Obtain .
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b)
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Obtain the unit tangent vector .
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c)
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Show that , thus verifying that is automatically a unit tangent vector.
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Example 2.3.4
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If is the position vector for the general parametric curve whose components are , and is arc length, show that is necessarily the unit (tangent) vector T.
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Example 2.3.5
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If is the position vector for , a curve parametrized by , the arc length, and is the unit tangent vector along , show that , thereby proving that the unit tangent vector is necessarily orthogonal to its derivative.
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Example 2.3.6
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If is a position vector and is its length, show that = .
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Example 2.3.7
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Given the plane curve defined by ,
a)
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Obtain , the radius-vector form of the curve, by the parametrization .
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b)
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Obtain , and , where and .
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c)
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Graph and the vectors and .
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d)
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Graph , and determine the point at which is a maximum in this interval.
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Example 2.3.8
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If is the position-vector form of the curve defined parametrically by the equations , ,
a)
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Obtain and , where and .
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b)
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Graph and the vectors .
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c)
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On the given interval, graph and determine its absolute minimum and the point on the curve where this minimum occurs.
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Example 2.3.9
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Given the two plane curves ,
a)
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At , obtain the equation of the line tangent to .
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b)
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Find the coordinates of the intersection of and the tangent line found in Part (a).
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c)
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Construct a vector from to the point found in Part (b).
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d)
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Obtain , the natural tangent vector at .
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e)
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Show that the vectors in Parts (c) and (d) are parallel.
(Hint: Show their components are proportional.)
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f)
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Draw both curves, the tangent line (Part (a)), and the tangent vector (Part (d)).
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