Cartesian coordinates in space (Sect. 12.1).

Transcription

1 Cartesian coordinates in space (Sect..). Overview of Multivariable Calculus. Cartesian coordinates in space. Right-handed, left-handed Cartesian coordinates. Distance formula between two points in space. Equation of a sphere. Overview of Multivariable Calculus Mth 3, Calculus I: f : R R, f (), differential calculus. Mth 33, Calculus II: f : R R, f (), integral calculus. Mth 34, Multivariable Calculus: f : R R, f (, ) } f : R 3 R, f (,, ) r : R R 3, r(t) = (t), (t), (t) scalar-valued. } vector-valued. We stud how to differentiate and integrate such functions.

2 The functions of Multivariable Calculus Eample An eample of a scalar-valued function of two variables, T : R R is the temperature T of a plane surface, sa a table. Each point (, ) on the table is associated with a number, its temperature T (, ). An eample of a scalar-valued function of three variables, T : R 3 R is the temperature T of an object, sa a room. Each point (,, ) in the room is associated with a number, its temperature T (,, ). An eample of a vector-valued function of one variable, r : R R 3, is the position function in time of a particle moving in space, sa a fl in a room. Each time t is associated with the position vector r(t) of the fl in the room. Cartesian coordinates in space (Sect..). Overview of vector calculus. Cartesian coordinates in space. Right-handed, left-handed Cartesian coordinates. Distance formula between two points in space. Equation of a sphere.

3 Cartesian coordinates. Cartesian coordinates on R : Ever point on a plane is labeled b an ordered pair (, ) b the rule given in the figure. (, ) Cartesian coordinates in R 3 : Ever point in space is labeled b an ordered triple (,, ) b the rule given in the figure. (,, ) Cartesian coordinates. Eample Sketch the set S = {,, = } R 3. Solution: = > > S

4 Cartesian coordinates. Eample Sketch the set S = {,, = } R 3. Solution: S Cartesian coordinates in space (Sect..). Overview of vector calculus. Cartesian coordinates in space. Right-handed, left-handed Cartesian coordinates. Distance formula between two points in space. Equation of a sphere.

5 Right and left handed Cartesian coordinates. Definition A Cartesian coordinate sstem is called right-handed (rh) iff it can be rotated into the coordinate sstem in the figure. (,, ) Right Handed Definition A Cartesian coordinate sstem is called left-handed (lh) iff it can be rotated into the coordinate sstem in the figure. (,, ) Left Handed No rotation transforms a rh into a lh sstem. Right and left handed Cartesian coordinates. Eample This coordinate sstem is right-handed. Eample This coordinate sstem is left handed.

6 Right and left handed Cartesian coordinates Remark: The same classification occurs in R : Right Handed Left Handed This classification is needed because: In R 3 we will define the cross product of vectors, and this product has different results in rh or lh Cartesian coordinates. There is no cross product in R. In class we use rh Cartesian coordinates. Cartesian coordinates in space (Sect..). Overview of vector calculus. Cartesian coordinates in space. Right-handed, left-handed Cartesian coordinates. Distance formula between two points in space. Equation of a sphere.

7 Distance formula between two points in space. Theorem The distance P P between the points P = (,, ) and P = (,, ) is given b P P = ( ) + ( ) + ( ). The distance between points in space is crucial to define the idea of limit to functions in space. Proof. Pthagoras Theorem. P P ( ) ( ) a ( ) P P = a + ( ), a = ( ) + ( ).

8 Distance formula between two points in space Eample Find the distance between P = (,, 3) and P = (3,, ). Solution: P P = (3 ) + ( ) + ( 3) = = 8 P P =. Distance formula between two points in space Eample Use the distance formula to determine whether three points in space are collinear. Solution: d P d 3 P d 3 P 3 P 3 d P d 3 P d 3 d + d 3 > d 3 d + d 3 = d 3 Not collinear, Collinear.

9 Cartesian coordinates in space (.) Overview of vector calculus. Cartesian coordinates in space. Right-handed, left-handed Cartesian coordinates. Distance formula between two points in space. Equation of a sphere. A sphere is a set of points at fied distance from a center. Definition A sphere centered at P = (,, ) of radius R is the set S = { P = (,, ) : P P } = R. R Remark: The point (,, ) belongs to the sphere S iff holds ( ) + ( ) + ( ) = R. ( iff means if and onl iff. )

10 An open ball is a set of points contained in a sphere. Definition An open ball centered at P = (,, ) of radius R is the set B = { P = (,, ) : P P } < R. Remark: The point (,, ) belongs to the open ball B iff holds ( ) + ( ) + ( ) < R. Eample Plot a sphere centered at P = (,, ) of radius R >. Solution: R

11 Eample Graph the sphere =. Solution: Complete the square. = [ = + + = + ( 4 ) + ( + 4 ) + 4. ( 4 ] ( 4 ) + ) = + ( + ) + =. Eample Graph the sphere =. Solution: Since = + ( + ) + =, we conclude that P = (,, ) and R =, therefore,

12 Eercise Given constants a, b, c, and d R, show that + + a b c = d is the equation of a sphere iff holds d > (a + b + c ). () Furthermore, show that if Eq. () is satisfied, then the epressions for the center P and the radius R of the sphere are given b P = (a, b, c), R = d + (a + b + c ). Vectors on a plane and in space (.) Vectors in R and R 3. Vector components in Cartesian coordinates. Magnitude of a vector and unit vectors. Addition and scalar multiplication.

13 Vectors in R and R 3. Definition A vector in R n, with n =, 3, is an ordered pair of points in R n, denoted as P P, where P, P R n. The point P is called the initial point and P is called the terminal point. P P P P Remarks: A vector in R or R 3 is an oriented line segment. A vector is drawn b an arrow pointing to the terminal point. A vector is denoted not onl b P P but also b an arrow over a letter, like v, or b a boldface letter, like v. Vectors in R and R 3. Remark: The order of the points determines the direction. For eample, the vectors P P and P P have opposite directions. P P P P P P P P Remark: B 85 it was realied that different phsical phenomena were described using a new concept at that time, called a vector. A vector was more than a number in the sense that it was needed more than a single number to specif it. Phenomena described using vectors included velocities, accelerations, forces, rotations, electric phenomena, magnetic phenomena, and heat transfer.

14 Vectors on a plane and in space (.) Vectors in R and R 3. Vector components in Cartesian coordinates. Magnitude of a vector and unit vectors. Addition and scalar multiplication. Components of a vector in Cartesian coordinates Theorem Given the points P = (, ), P = (, ) R, the vector P P determines a unique ordered pair denoted as follows, P P = ( ), ( ). Proof: Draw the vector P P in Cartesian coordinates. P P P ( ) ( ) P Remark: A similar result holds for vectors in space.

15 Components of a vector in Cartesian coordinates Theorem Given the points P = (,, ), P = (,, ) R 3, the vector P P determines a unique ordered triple denoted as follows, P P = ( ), ( ), ( ). Proof: Draw the vector P P in Cartesian coordinates. ( ) P P P P ( ) ( ) Components of a vector in Cartesian coordinates Eample Find the components of a vector with initial point P = (,, 3) and terminal point P = (3,, ). Solution: P P = (3 ), ( ( )), ( 3) P P =, 3,. Eample Find the components of a vector with initial point P 3 = (3,, 4) and terminal point P 4 = (5, 4, 3). Solution: P 3 P 4 = (5 3), (4 ), (3 4) P 3 P 4 =, 3,. Remark: P P and P 3 P 4 have the same components although the are different vectors.

16 Components of a vector in Cartesian coordinates Remark: The vector components do not determine a unique vector. The vectors u, v and P have the same components but the are all different, since the have different initial and terminal points. P v u P v v v v Definition Given a vector P P = v, v, the standard position vector is the vector P, where the point = (, ) is the origin of the Cartesian coordinates and the point P = (v, v ). Components of a vector in Cartesian coordinates Remark: Vectors are used to describe motion of particles. The position r(t), velocit v(t), and acceleration a(t) at the time t of a moving particle are described b vectors in space. a(t) r(t) v(t) r()

17 Vectors on a plane and in space (.) Vectors in R and R 3. Vector components in Cartesian coordinates. Magnitude of a vector and unit vectors. Addition and scalar multiplication. Magnitude of a vector and unit vectors. Definition The magnitude or length of a vector P P is the distance from the initial point to the terminal point. If the vector P P has components P P = ( ), ( ), ( ), then its magnitude, denoted as P P, is given b P P = ( ) + ( ) + ( ). If the vector v has components v = v, v, v, then its magnitude, denoted as v, is given b v = v + v + v.

18 Magnitude of a vector and unit vectors. Eample Find the length of a vector with initial point P = (,, 3) and terminal point P = (4, 3, ). Solution: First find the component of the vector P P, that is, P P = (4 ), (3 ), ( 3) P P = 3,,. Therefore, its length is P P = ( ) P P =. Eample If the vector v represents the velocit of a moving particle, then its length v represents the speed of the particle. Magnitude of a vector and unit vectors. Definition A vector v is a unit vector iff v has length one, that is, v =. Eample Show that v = 4, 4, 3 4 is a unit vector. Solution: v = = 4 v =. Eample The unit vectors i =,,, j =,,, and k =,, are useful to epress an other vector in R 3. k i j

19 Vectors on a plane and in space (.) Vectors in R and R 3. Vector components in Cartesian coordinates. Magnitude of a vector and unit vectors. Addition and scalar multiplication. Addition and scalar multiplication. Definition Given the vectors v = v, v, v, w = w, w, w in R 3, and a number a R, then the vector addition, v + w, and the scalar multiplication, av, are given b Remarks: v + w = (v + w ), (v + w ), (v + w ), av = av, av, av. The vector v = ( )v is called the opposite of vector v. The difference of two vectors is the addition of one vector and the opposite of the other vector, that is, v w = v + ( )w. This equation in components is v w = (v w ), (v w ), (v w ).

20 Addition and scalar multiplication. Remark: The addition of two vectors is equivalent to the parallelogram law: The vector v + w is the diagonal of the parallelogram formed b vectors v and w when the are in their standard position. V V+W V v (v+w) W w (v+w) w v Addition and scalar multiplication. Remark: The addition and difference of two vectors. v v w v+w w Remark: The scalar multiplication stretches a vector if a > and compresses the vector if < a <. V a V a> a V < a < V a =

21 Addition and scalar multiplication. Eample Given the vectors v =, 3 and w =,, find the magnitude of the vectors v + w and v w. Solution: We first compute the components of v + w, that is, v + w = ( ), (3 + ) v + w =, 5. Therefore, its magnitude is v + w = + 5 v + w = 6. A similar calculation can be done for v w, that is, v w = ( + ), (3 ) v w = 3,. Therefore, its magnitude is v w = 3 + v w =. Addition and scalar multiplication. Theorem If the vector v, then the vector u = v v is a unit vector. Proof: (Case v R onl). If v = v, v R, then v = This is a unit vector, since v + v, and u = v v = v v, v. v u = v = v ( v ) ( v ) + = v v v v + v = v v =.

22 Addition and scalar multiplication. Theorem Ever vector v = v, v, v in R 3 can be epressed in a unique wa as a linear combination of vectors i =,,, j =,,,and k =,, as follows v = v i + v j + v k. Proof: Use the definitions of vector addition and scalar multiplication as follows, v = v, v, v = v,, +, v, +,, v = v,, + v,, + v,, k i j v = v i + v j + v k. Addition and scalar multiplication. Eample Epress the vector with initial and terminal points P = (,, 3), P = (, 4, 5) in the form v = v i + v j + v k. Solution: First compute the components of v = P P, that is, v = ( ), (4 ), (5 3) =, 4,. Then, v = i + 4j + k. Eample Find a unit vector w opposite to v found above. Solution: Since v = ( ) = = 4, we conclude that w =, 4,. 4