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Chapter 11: Q.49 (page 860)

Show that the graph of the vector function $r (t) = ⟨ 3 \sin t, 5 \cos t, 4 \sin t ⟩$ is a circle. (Hint: Show that the graph lies on a sphere and in a plane.)

Short Answer

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Step by step solution

Step 1. Given information.

given,

$r (t) = ⟨ 3 \sin t, 5 \cos t, 4 \sin t ⟩$

Step 2.

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Most popular questions from this chapter

Prove that the tangent vector is always orthogonal to the position vector for the vector-valued function.

$r (t) = <\sin t, \sin t \cos t, \cos^{2} t>$

In Exercises 19–21 sketch the graph of a vector-valued function $r (t) = (x (t), y (t))$ with the specified properties. Be sure to indicate the direction of increasing values oft.

Domainlocalid="1649578783745" $t \geq 0, r (0) = (1, 0), r (2) = (0, - 1), \lim_{t \to \infty} x (t) = - 2, and \lim_{t \to \infty} y (t) = - 3 .$

The DNA molecule takes the shape of a double helix—two helices that stay a roughly uniform distance apart.

(a) Neglecting actual dimensions, we can model one strand of DNA using the vector function $h_{1} (t) = ⟨ \cos (t), \sin (t), α t ⟩ .$ .

Sketch the graph of $h_{1}$ . What is the effect of the parameter $α$ ?

(b) The second strand of DNA can be constructed by shifting the first. Does the graph of $h_{2} (t) = ⟨ \cos (t), \sin (t), α t + β ⟩$ ever intersect that of $h_{1}$ ?

(c) The distance between two curves is the minimum distance between any two points on the curves. What is the distance between $h_{1}$ and $h_{2}$ if $α = 1 a n d β = π$ ? (Hint: Write two points on the curves using parameters $t_{1}$ and $t_{2}$ , expand the formula for the distance between them, and then use a trigonometric identity for addition. Then let

$s = t_{1} - t_{2}$ and minimize.).

Evaluate and simplify the indicated quantities in Exercises 35–41.
$(1, t, t^{2}) - (t, t^{2}, t^{3})$

Let $r (t) = (x (t), y (t)), t \in [a, \infty),$ be a vector-valued function, where a is a real number. Under what conditions would the graph of r have a horizontal asymptote as $t \to \infty ?$ Provide an example illustrating your answer.

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Show that the graph of the vector function r(t)=⟨3sin⁡t,5cos⁡t,4sin⁡t⟩is a circle. (Hint: Show that the graph lies on a sphere and in a plane.)

Short Answer

Step by step solution

Step 1. Given information.

Step 2.

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Show that the graph of the vector function $r (t) = ⟨ 3 \sin t, 5 \cos t, 4 \sin t ⟩$ is a circle. (Hint: Show that the graph lies on a sphere and in a plane.)