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Chapter 11: Q. 1TF (page 891)

A decomposition of the acceleration vector: Find ${comp}_{v (t)} a (t),$ wherev and a are the velocity and acceleration vectors, respectively, of the following functions.
$r (t) = <t, t^{2}, t^{3}>$

Short Answer

Expert verified

The component of a(t)along v(t)is ${comp}_{v (t)} a (t) = \frac{4 t + 18 t^{3}}{\sqrt{1 + 4 t^{2} + 9 t^{4}}} .$

Step by step solution

01

Step 1. Given Information.

The given function is $r (t) = <t, t^{2}, t^{3}> .$

02

Step 2. Calculating the velocity and acceleration vector.

Let's calculate the given function,

$r (t) = <t, t^{2}, t^{3}> r' (t) = v (t) = <1, 2 t, 3 t^{2}> r'' (t) = a (t) = <0, 2, 6 t> Now, ||v (t)|| = \sqrt{{(1)}^{2} + {(2 t)}^{2} + {(3 t^{2})}^{2}} = \sqrt{1 + 4 t^{2} + 9 t^{4}} . . . . . . . . . . (a) Let' s find (a (t) 路 v (t)) = (0, 2, 6 t) 路 (1, 2 t, 3 t^{2}) = 4 t + 18 t^{3} . . . . . . . . . . (b)$

03

Step 3. Solve.

Put (a) and (b) in $c o m p_{v (t)} a (t) = \frac{|a (t) . v (t)|}{||v (t)||},$

$c o m p_{v (t)} a (t) = \frac{|4 t + 18 t^{3}|}{\sqrt{1 + 4 t^{2} + 9 t^{4}}} c o m p_{v (t)} a (t) = \frac{4 t + 18 t^{3}}{\sqrt{1 + 4 t^{2} + 9 t^{4}}}$

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Most popular questions from this chapter

Prove that the cross product of two orthogonal unit vectors is a unit vector.

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

Given a vector-valued function r(t) with domain $鈩�,$ what is the relationship between the graph of r(t) and the graph of r(kt), where k > 1 is a scalar?

Let Cbe the graph of a vector-valued function r. The plane determined by the vectors $N (t_{0}), B (t_{0})$ and containing the point $r (t_{0})$ is called the normal plane forC at $r (t_{0})$ . Find the equation of the normal plane to the helix determined by $r (t) = < \cos t, \sin t, t >$ for $t = 蟺$ .

For each of the vector-valued functions, find the unit tangent vector.

r (t) = (\cos (伪 t), \sin (伪 t))

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