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.
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).
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.
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
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
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
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
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
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
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
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.
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).