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Answer: Properties of space curves Do the following

Chapter 12, Problem 37RE

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QUESTION:

Properties of space curves  Do the following calculations for all values of t for which the given curve is defined.

(a) Find the tangent vector and the unit tangent vector.

(b) Find the curvature.

(c) Find the principal unit normal vector.

(d) Verify that |N|=1 and \(\mathbf{T} \cdot \mathbf{N}=0\).

(e) Graph the curve and sketch T and N at two points.

\(\mathbf{r}(t)=\cos t \mathbf{i}+2 \cos t \mathbf{j}+\sqrt{5} \sin t \mathbf{k}\), for \(0 \leq t \leq 2 \pi\)

Questions & Answers

QUESTION:

Properties of space curves  Do the following calculations for all values of t for which the given curve is defined.

(a) Find the tangent vector and the unit tangent vector.

(b) Find the curvature.

(c) Find the principal unit normal vector.

(d) Verify that |N|=1 and \(\mathbf{T} \cdot \mathbf{N}=0\).

(e) Graph the curve and sketch T and N at two points.

\(\mathbf{r}(t)=\cos t \mathbf{i}+2 \cos t \mathbf{j}+\sqrt{5} \sin t \mathbf{k}\), for \(0 \leq t \leq 2 \pi\)

ANSWER:

Solution 37RE

a. The unit tangent vector is T (t) = (-sin ti - 2sin tj + cos tk)

b. The curvature is K=

c. The principal unit normal vector is

N =.

 N= ((-cos

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