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Chapter 11: Q. 39 (page 860)

Evaluate and simplify the indicated quantities in Exercises 35–41.
$(\sin t, \cos t) . (\cos t, - \sin t)$

Short Answer

Expert verified

The simplification of $(\sin t, \cos t) . (\cos t, - \sin t)$ is $0$ .

Step by step solution

Step 1. Given Information.

The quantities:

$(\sin t, \cos t) . (\cos t, - \sin t)$

Step 2. Do the dot product.

By Dot product,

$(\sin t, \cos t) . (\cos t, - \sin t) = \sin t . \cos t + (- \sin t . \cos t) = 0$

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Most popular questions from this chapter

Using the definitions of the normal plane and rectifying plane in Exercises 20 and 21, respectively, find the equations of these planes at the specified points for the vector functions in Exercises 40–42. Note: These are the same functions as in Exercises 35, 37, and 39.

$r (t) = <e^{t}, e^{- t}, \sqrt{2} t) at t = 0$

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

$r (t) = (\sin 2 t, \cos 2 t, t)$

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r (t) = (\cos^{2} t, 4 i n t, t) f o r t \in [0, 2 π]$

Explain why we do not need an “epsilon–delta” definition for the limit of a vector-valued function.

Let $r (t) = ⟨ x (t), y (t) |$ be a vector-valued function defined on an open interval containing the point $t_{0}$ . Prove that r(t) is continuous at $t_{0}$ if and only if $x (t)$ and $y (t)$ are both continuous at $t_{0}$ .

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Evaluate and simplify the indicated quantities in Exercises 35–41.(sint,cost).(cost,-sint)

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Do the dot product.

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Most popular questions from this chapter

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Evaluate and simplify the indicated quantities in Exercises 35–41.
$(\sin t, \cos t) . (\cos t, - \sin t)$