Explanation Given data: Vector function is r ( t ) = t 2 i + t 4 j + t 6 k . Definition: Consider a vector function as r ( t ) = f ( t ) i + g ( t ) j + h ( t ) k , then parametric equations to plot space curve C are, x = f ( t ) y = g ( t ) z = h ( t ) Here, f ( t ) , g ( t ) , and h ( t ) are component functions of r , and x , y , and z are parametric equations of C . From definition, write the parametric equations of vector function r ( t ) = t 2 i + t 4 j + t 6 k . x = t 2 y = t 4 (1) z = t 6 (2) From parametric equations, it is obvious that, for any value of t when t ≠ 0 the parametric equations are positive whereas, if t = 0 the values of parametric equations are equal to zero

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Chapter 13.1, Problem 15E

To determine

To sketch: the curve of vector equation r(t)=t2i+t4j+t6k and indicate the direction of increment of t.

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Chapter 13.1, Problem 14E

Chapter 13.1, Problem 16E

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1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps (each step must be justified). Theorem 0.1 (Abel's Theorem). If y1 and y2 are solutions of the differential equation y" + p(t) y′ + q(t) y = 0, where p and q are continuous on an open interval, then the Wronskian is given by W (¥1, v2)(t) = c exp(− [p(t) dt), where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or W (y1, y2)(t) = 0 for every t in I. 1. (a) From the two equations (which follow from the hypotheses), show that y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0, 2. (b) Observe that Hence, conclude that (YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0. W'(y1, y2)(t) = yY2 - Y1 y2- W' + p(t) W = 0. 3. (c) Use the result from the previous step to complete the proof of the theorem.

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Question 4 Find an equation of (a) The plane through the point (2, 0, 1) and perpendicular to the line x = y=2-t, z=3+4t. 3t, (b) The plane through the point (3, −2, 8) and parallel to the plane z = x+y. (c) The plane that contains the line x = 1+t, y = 2 − t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1. (d) The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1+t, and z = 2-t. (e) The plane that contains the lines L₁: x = 1 + t, y = 1 − t, z = 2t and L2 : x = 2 − s, y = s, z = 2.

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the curve of vector equation r ( t ) = t 2 i + t 4 j + t 6 k and indicate the direction of increment of t .

Solution Summary: The author illustrates how the vector function r(t) is sketched and the increasing direction of t is indicated with arrow.

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To sketch: the curve of vector equation r(t)=t2i+t4j+t6k and indicate the direction of increment of t.

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Chapter 13 Solutions