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Chapter 11: Q. 16 (page 879)

Given a twice-differentiable vector-valued function $r (t)$ and a point $t_{0}$ in its domain, what is the osculating plane at $r (t_{0})$ ?

Short Answer

Expert verified

The osculating plane at $B (t_{0}) = T (t_{0}) 脳 N (t_{0})$ .

Step by step solution

01

Step 1. Given information.

Consider the given question,

A twice-differentiable vector-valued function is $r (t_{0})$ is any point in the domain of $r (t)$ .

02

Step 2. Definition of osculating plane at rt0.

Assume $r (t) = <x (t), y (t), z (t)>$ be a differentiable vector function on the interval $I 鈯� R$ such that,

$T' (t_{0}) 鈮� 0$ where, $t_{0} 鈭� I$ . Then the osculating plane at $r (t_{0})$ is defined as,

$B (t_{0}) . <x - x (t_{0}), y - y (t_{0}), z - z (t_{0})> = 0$

Where, $B (t_{0}) = T (t_{0}) 脳 N (t_{0})$ .

$T (t_{0}), N (t_{0})$ are the orthogonal unit vectors. so their cross product $B (t_{0})$ is another unit vector and is orthogonal to both $T (t_{0}), N (t_{0})$ .

The osculating plane contains the point $r (t_{0})$ and has $B (t_{0})$ as its normal vector.

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Most popular questions from this chapter

Let $r_{1} (t)$ and $r_{2} (t)$ both be differentiable three-component vector functions. Prove that

\frac{d}{d t} (r_{1} (t) 脳 r_{2} (t)) = r_{1}^{'} (t) 脳 r_{2} (t) + r_{1} (t) 脳 r_{2}^{'} (t)

(This is Theorem 11.11 (d).)

If $伪$ , $尾$ , and $纬$ are nonzero constants, the graph of a vector function of the formrole="math" localid="1649577570077" $r (t) = <伪 t, 尾 t^{2}, 纬 t^{3}>$ is called a twisted cubic. Prove that a twisted cubic intersects any plane in at most three points.

Annie is conscious of tidal currents when she is sea kayaking. This activity can be tricky in an area south-southwest of Cattle Point on San Juan Island in Washington State. Annie is planning a trip through that area and finds that the velocity of the current changes with time and can be expressed by the vector function

$<0.4 \cos 鈦� (\frac{蟺 (t 鈭� 8)}{6}), 鈭� 1.1 \cos 鈦� (\frac{蟺 (t 鈭� 11)}{6})>,$

where t is measured in hours after midnight, speeds are given in knots and point due north.

(a) What is the velocity of the current at 8:00 a.m.?

(b) What is the velocity of the current at 11:00 a.m.?

(c) Annie needs to paddle through here heading southeast, 135 degrees from north. She wants the current to push her. What is the best time for her to pass this point? (Hint: Find the dot product of the given vector function with a vector in the direction of Annie鈥檚 travel, and determine when the result is maximized.)

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

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

Evaluate the limits in Exercises 42鈥�45.

$\underset{t 鈫� 0}{l i m} (\sin 3 t, \cos 3 t)$

See all solutions

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