Math 11 Fall 2016 Section 1 Monday, September 19, Definition: A vector parametric equation for the line parallel to vector v = x v, y v, z v

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1 Math Fall 06 Section Monay, September 9, 06 First, some important points from the last class: Definition: A vector parametric equation for the line parallel to vector v = x v, y v, z v passing through the point (x 0, y 0, z 0 ) with position vector r 0 = x 0, y 0, z 0 is Scalar parametric equations for this line are r = r 0 + t v, or x, y, z = x 0, y 0, z 0 + t x v, y v, z v. x = x 0 + tx v y = y 0 + ty v z = z 0 + tz v. Definition: A vector equation for the plane perpenicular to the vector n = a, b, c containing the point (x 0, y 0, z 0 ) with position vector r 0 = x 0, y 0, z 0 is The scalar (linear) equation is: n ( r r 0 ) = 0 ax + by + cz = ax 0 + by 0 + cz 0 Note: From a linear equation ax+by +cz = for a plane, you can rea off the normal vector n = a, b, c. Definition: Planes are calle parallel if they have parallel normal vectors. The angle between two planes is the angle between their normal vectors.

2 Last time we saw that if an object starts at point (a, b, c) with position vector r 0 an moves with constant velocity v = v x, v y, v z for t secons, then its final position is given by (a + v x t, b + v y t, c + v z t). We can express this by a function that takes the elapse time t to the position vector after t secons: f(t) = a + v x t, b + v y t, c + v z t = a, b, c + t v x, v y, v z, f(t) position = r 0 initial position + t time v velocity isplacement where t represents time in secons, t = 0 is the starting time, an f(t) is the object s position vector at time t. The omain of f is the real number line R an the range of f is a line containe in three-imensional space R Suppose another object is traveling counterclockwise aroun the unit circle x + y = in the plane R. At time t = 0 it is at the point (, 0), an it travels at constant spee, making one complete trip aroun the circle in π units of time.. What is the angle between the object s position vector an the positive x-axis when t = π? What is the object s position at that time? π an (0, ), respectively. (When t = π it has complete one trip aroun the circle, through an angle of π, so when t = π it has gone a quarter of the way aroun.). At what time t > 0 is the angle between the object s position vector an the positive x-axis first equal to 4π?, t = 4π. What is the angle θ(t) between the object s position vector an the positive x-axis at time t? θ(t) = t. 4. What is the object s position at time t? (cos(t), sin(t)). In this case, too, we can write the object s position as a function of time: f(t) = f(t) = cos(t), sin(t).

3 A vector value function r(t) is a function that takes a real number t to a vector r(t). If the range consists of vectors in R, for example, we write r : R R. We may say r maps R to R. r function : R contains omain R contains range Example: If r(t) = cos(t), sin(t), then r : R R. The omain of r is R an the range of r is the unit circle in R Definition: If a curve γ is the range of a vector function r, we say that r parametrizes γ, or is a parametrization of γ. (Think of t in r(t) as a parameter ifferent values of t give ifferent points on γ. You can also think r(t) is the position at time t of a point moving along γ.) For example, r(t) = r 0 +t v parametrizes a line, an r(t) = cos(t), sin(t), for 0 t π, parametrizes the unit circle in R. Sometimes we speak of oriente curves, which are curves with a chosen irection of motion. For example, the unit circle gives rise to two oriente curves, one oriente clockwise an one oriente counterclockwise. The function r(t) = cos(t), sin(t) 0 t π parametrizes the unit circle with counterclockwise orientation. The function r(t) = ( t) r 0 + t r 0 t parametrizes the line segment from r 0 to r, oriente in that irection. Example: Give a parametrization of the curve in R that is the intersection of the surface z = x y an the cyliner x + y =. The projection on the xy-plane is the unit circle x + y =, so we can set x = cos(t) an y = sin(t). Then z = x y = cos (t) sin (t) = cos(t). f(t) = cos(t), sin(t), cos(t). We can take 0 t π, if we want to trace out the entire curve without retracing any part.

4 We prefer our parametrizations to be continuous. You may recall that for f : R R we say lim f(x) = L means for every ε > 0 [esire output accuracy] there is a δ > 0 x a [require input accuracy] such that, for every x, x a < δ & x a within input accuracy of a, but not a = f(x) L < ε. within output accuracy of L If r : R R n, we say something very similar lim r(t) = L means for every ε > 0 [esire output accuracy] there is a δ > 0 t a [require input accuracy] such that, for every t, t a < δ & t a within input accuracy of a, but not a = istance between r(t) an L {}}{ r(t) L < ε. within output accuracy of L In practice, we on t often use this formal efinition of limit to compute limits, although we may use it to prove things. then Theorem: If Example: r(t) = r x (t), r y (t), r z (t), lim r(t) = lim r x (t), lim r y (t), lim r z (t). t a t a t a t a sin(t) lim t 0 t lim sin(t) t 0 t, t ln(t ) =, lim t ln(t ) = t 0 (using l Hôpital s rule) cos(t) ln(t ) lim, lim t 0 t 0 = t t t, lim t 0 = t, 0. Definition: A vector function r(t) is continuous at a if lim t a r(t) = r(a). 4

5 Example: Give parametrizations of the following curves:. The intersection of the paraboloi z = x + y an the plane x =.. The ellipse x + 4y = 4 in R. Example: Sketch an/or escribe the curves parametrize by the following functions:. r(t) = cos(t), sin(t), t.. r(t) = t, t, t. 5

6 We etermine that if an object travels with constant velocity v for time perio t, its isplacement is = t v. We can use this the other way: If an object travels with constant velocity for time perio t an its isplacement is, then its velocity is v = t. If the velocity is not constant, we say t is the average velocity. Example: An object travels aroun the unit circle in the xy-plane, an its position at time t is the point (cos t, sin t) (with time an istance measure in your favorite units). The object s isplacement between times t an t + t is cos(t + t), sin(t + t) cos(t), sin(t) = cos(t + t) cos(t), sin(t + t) sin(t) If t is small enough, it is not a ba approximation to suppose that between times t an t, the object is traveling at constant velocity. The object s velocity between times t an t is approximately cos(t + t) cos(t), sin(t + t) sin(t) = t cos(t + t) cos(t) sin(t + t) sin(t), t t The object s instantaneous velocity at time t is (letting t 0) cos(t + t) cos(t) sin(t + t) sin(t) lim, = t 0 t t cos(t + t) cos(t) sin(t + t) sin(t) lim, lim = sin(t), cos(t). t 0 t t 0 t 6

7 then Definition: The erivative of a vector function is efine by Theorem: If r(t) = lim ( r(t + t) r(t)). t t 0 t r(t) = r x (t), r y (t), r z (t), t r(t) = t r x(t), t r y(t), t r z(t). Example: cos(t), sin(t) = sin(t), cos(t). t If r(t) enotes the position of a moving object at time t, then the erivative r (t) enotes the velocity of that object, its magnitue r (t) is the object s spee, an the unit vector T (t) in the irection of r (t) gives the irection of motion. The unit vector T (t) is tangent to the object s path, an is calle the unit tangent vector. 7

8 Example: An object moves in the plane with position function r(t) = cos(t), sin(t). Fin a vector parametric equation for the line tangent to the object s path at the point,., line. ( π ) = r is the object s position at time t = π. This is a point on the tangent r (t) = sin(t), cos(t) is the object s velocity at time t. ( π ) r =, is the object s velocity at time t = π. Since velocity points in the irection of motion, this is a vector in the irection of the tangent line. r =, + t, is an equation for the tangent line. The function l(t) =, + t, parametrizes the tangent line. Notice, the function l(t) represents the motion of an object traveling along the tangent line at the same velocity our original object has at time t = π, starting at time t = 0 at the position of our original object at time t = π. If we want the object with position function l to be at the same point as our original object at time t = π, we shoul ajust its clock. We can o this by replacing t with t π, to get a new position function f(t) =, + ( t π ),. You can check this function has the same value an erivative as r(t) when t = π. 8

9 Theorem: If all functions mentione are ifferentiable, then t (a r(t) + b p(t)) = a r (t) + b p (t) t ( r(f(t)) = f (t) r (f(t)) t (f(t) r(t)) = f (t) r(t) + f(t) r (t) t ( r(t) p(t)) = r (t) p(t) + r(t) p (t) t ( r(t) p(t)) = r (t) p(t) + r(t) p (t) Theorem: If r(t) is ifferentiable, then r(t) is constant if an only if r(t) r (t) for all t. Proof: We know r(t) is constant if an only if r(t) is constant, an r(t) = r(t) r(t). A function is constant if an only if its erivative is always zero, so r(t) is constant if an only if 0 = t ( r(t) r(t)) = r (t) r(t) + r(t) r (t) = r(t) r (t). We also know that r(t) r (t) = 0 if an only if r(t) r (t). Putting all this together, r(t) is constant if an only if r(t) r (t) for all t. Definition: If a curve γ is the range of a vector function r, we say that r parametrizes γ, or is a parametrization of γ. (Think of t in r(t) as a parameter ifferent values of t give ifferent points on γ.) If r (t) is efine an nonzero for every t, then r is a regular parametrization, or smooth parametrization, of γ. A curve with a regular parametrization is calle a smooth curve. Definition: r x (t), r y (t), r z (t) t = r x (t) t, r y (t) t, r z (t) t Theorem b a b r x (t), r y (t), r z (t) t = r x (t) t, b a a b r (t) t = r(b) r(a). a r y (t) t, If r(t) is the position of a moving object at time t then this is: the net isplacement between times t = a an t = b. b a r z (t) t 9

10 Example: The position function of a moving object is r(t) =, cos(t), sin(t).. Describe the object s path as completely as possible.. Fin the object s velocity in general an at time t = π.. Fin an equation for the line tangent to the object s path at the point, 0,. 4. Fin π 0 r (t) t. (What oes this represent?) 5. Fin the istance between the object s position at t = 0 an the object s position at t = π. (How is this relate to your answer to part (4)?) 6. Using part (), fin the length of the curve the object travels along between time t = 0 an time t = π. (Don t try to use calculus for this one. Use geometry.) 0

11 Example: If the position of a moving object at time t is r(t) = cos(t), sin(t), t, what can we say about the motion of that object? Once you have sai everything else you have to say: What is the projection of the object s path on the xy-plane? What is the angle between the object s velocity vector at time t an the xy-plane? (Think about this one. It requires some cleverness.) Given that, can you make a guess about the length of the path the object travels along between t = 0 an t = π? We will soon learn how to compute lengths of paths.

12 Challenge: Give parametrizations of the following curves:. The portion of the intersection of the sphere x + y + z = 4 with the plane x = y + for which x 0 an y 0.. The intersection of the plane z = an the surface z = x 4 + y 4. Challenge: Sketch an/or escribe the curves parametrize by the following functions:. r(t) = cos(t) an,, +sin(t),, 0,, 0. You may notice that,, are unit vectors an are perpenicular to each other.. The curve that lies in the cone z = x + y an whose projection onto the xy-plane is parametrize by r(t) = t cos(t), t sin(t).