Lectre 3 The dot prodct Dan Nichols nichols@math.mass.ed MATH 33, Spring 018 Uniersity of Massachsetts Janary 30, 018 () Last time: 3D space Right-hand rle, the three coordinate planes 3D coordinate system: (x, y, z) Eqations of certain planes and of spheres
(3) The dot prodct We e learned how to add/sbtract two ectors and how to mltiply a ector by a scalar There s another crcial operation we can do with ectors: the dot prodct. The dot prodct mltiplies two ectors and gies yo a scalar, NOT another ector. Definition Let = 1,, 3 and = 1,, 3 be ectors. The dot prodct of and is = 1 1 + + 3 3. (4) The dot prodct = 1 1 + + 3 3 In other words: mltiply each corresponding pair of components, then add all these prodcts together. Also called the scalar prodct or inner prodct. Defined exactly the same in D space: Or in any other nmber of dimensions. a 1, a b 1, b = a 1 b 1 + a b a 1, a, a 3, a 4, a 5 b 1, b, b 3, b 4, b 5 = a 1 b 1 + a b + a 3 b 3 + a 4 b 4 + a 5 b 5
(5) The dot prodct: example Example 1: Let s calclate the dot prodct of these two ectors. = 0, 1, 5 = 1, 3, 1 = 0 ( 1) + 1 3 + 5 ( 1) = 0 + 3 5 = (6) The dot prodct Q: Why call it a prodct when it isn t really the same as mltiplication? If we mltiply two things, we expect to get another thing of the same type If we mltiply two nonzero things, we expect the answer will neer be zero The dot prodct breaks both of these rles! A: it follows most of the other rles of mltiplication. ( + w) = + w (Distribtie property) = (Commtatie property) 0 = 0 (c) (d) = cd( ) (Compatible with scalar mltiplication)
(7) The dot prodct The dot prodct makes sense becase 1 It satisfies these nice algebraic properties (distribtiity, etc) It has an important geometric meaning (more on this later) Other possible ways to define mltiplication on ectors don t make as mch sense. For instance, component-wise mltiplication: = 1 1,, 3 3 The operation does not hae these algebraic properties, no geometric meaning. Mostly seless. (8) The dot prodct: proof of distribtie property Sppose we hae three ectors: = 1,, 3, = 1,, 3, and w = w 1, w, w 3. ( + w) = 1,, 3 1 + w 1, + w, 3 + w 3 = 1 ( 1 + w 1 ) + ( + w ) + 3 ( 3 + w 3 ) = 1 1 + 1 w 1 + + w + 3 3 + 3 w 3 = ( 1 1 + + 3 3 ) + ( 1 w 1 + w + 3 w 3 ) = + w
(9) The dot prodct: proof of compatibility with scalar mltiplication Let = 1,, 3 and = 1,, 3 be ectors. (c) (d) = (c 1,, 3 ) (d 1,, 3 ) = c 1, c, c 3 d 1, d, d 3 = (c 1 )(d 1 ) + (c )(d ) + (c 3 )(d 3 ) = (cd)( 1 1 ) + (cd)( 3 ) + (cd)( 3 3 ) = (cd) ( 1 1 + + 3 3 ) = (cd)( ) (10) The dot prodct: more examples Example : Let = 1, 0, 4, let =,, 7, and let w = 0, 0, 3. Compte the following: = 1, 0, 4,, 7 =1( ) + 0( ) + 4 7 = + 0 + 8 =6 (3) =3( ) =3 6 =78 ( w) = ( w) =6 ( 1, 0, 4 0, 0, 3 ) =6 (1) =
(11) The dot prodct: geometric interpretation Vector addition and scalar mltiplication can be nderstood geometrically (withot relying on components) What is the geometric interpretation of the dot prodct? In other words, what does the dot prodct mean in terms of jst arrows withot knowing explicit components? (1) The dot prodct: geometric interpretation Here s something interesting: = 1,, 3 1,, 3 = 1 + + 3 1,, 3 = ( ) 1 + + 3 = 1 + + 3 How does the dot prodct work with the standard basis ectors? Reminder: i = 1, 0, 0, j = 0, 1, 0, k = 0, 0, 1 i i = j j = k k = 1 i j = i k = j k = 0
(13) The dot prodct: geometric interpretation In fact, wheneer two ectors are perpendiclar (or orthogonal), their dot prodct is 0. y = 3,, = 4, 6 = 3 4 + 6 = 1 + 1 = 0 x Rle: = 0 if and only if and are orthogonal. This is the best way to check orthogonality. (14) The dot prodct: parallel and orthogonal ectors Here s how we define the words orthogonal and parallel : and are orthogonal (perpendiclar) if = 0. and are parallel if = c for some scalar c. These definitions make sense in any nmber of dimensions.
(15) The dot prodct: angle form Alternate definition: = cos θ, where θ is the angle between and. y θ x (16) The dot prodct: angle form Let s proe this in the special case where = i and = 1. y = 1, 0, = cos θ, sin θ = 1 cos θ + 0 sin θ = cos θ Remember that (c) (d) = (cd)( ). So een if and are not nit ectors, = cos θ. (We e still only proed it for = ci... ) r = 1 θ x How to proe it for arbitrary,? (withot assming = ci) Easy way: linear algebra rotation matrices Hard way: trigonometric identities. Try it if yo re bored! Easiest way is sing the law of cosines, bt that s cheating
(17) The dot prodct: angle between ectors Writing the dot prodct in angle form makes the geometric meaning clear. = cos θ Bigger for two ectors that point more in the same direction. Smaller for ectors that are closer to orthogonal Bigger for longer ectors. Smaller for shorter ectors. Yo can se the dot prodct to find θ easily! cos θ = ( ) θ = arccos (18) The dot prodct: angle between ectors So the dot prodct measres how mch two ectors agree with each other, weighted by their length. > 0 if θ < 90 = 0 if θ = 90 (orthogonal) < 0 if θ > 90 y y y x x x
(19) The dot prodct: angle example Example 3: Find the angle θ between the ectors = 1, 3, 1 and =,, 3. z = 11 = 17 = 7 ( ) 7 θ = arccos.108 11 17 y x (0) Vector and scalar projections Sometimes we want to compare two ectors asymmetrically. We want to know how mch of lies in the same direction as? Definition Gien ectors and, we can always write as a sm of two ectors = + where one part is parallel to and the other part is orthogonal to. We call the parallel part the ector projection of onto. = proj = proj
(1) Vector and scalar projections There s an easy way to calclate proj sing the dot prodct. First, let s figre ot what the magnitde of proj mst be. Definition The scalar projection of onto, denoted comp, is the signed magnitde of the ector projection. This is sometimes called the component of along. comp = The scalar projection is closely related to the dot prodct measres how mch the ectors agree, weighted by both magnitdes comp measres how mch the ectors agree, ignoring (asymmetrical) () Vector and scalar projections Now we know two things abot proj : 1. Its magnitde is comp =.. Its direction is the nit ector û = ( 1 Therefore, ). proj = (comp ) û ( ) ( ) 1 = ( ) = This is an easy formla for calclating ector projections.
(3) Vector and scalar projections The dot prodct and the scalar projection are similar. Here s how to distingish them: Dot prodct measres how mch and mtally agree with each other. Scalar projection comp measres how mch agrees with. Scalar projection comp measres how mch agrees with. (4) Vector and scalar projections: example Example 4: Let = 1, 1 and let = 4, 5. Find comp and proj. y comp = = 4 1 + 5 1 ( 1) + 1 = 1. proj x proj = (comp ) ( ) comp = = 1/ 1, 1 = 1/, 1/.
(5) Applications: work Sppose yo moe an object from point A to point B by applying a constant force F. F B A The work done in this sitation is the energy spent to moe the object: W = F d d = AB is the object s displacement ector F is the net force ector on the object as it s moing W is the work, a scalar. Often measred in newton-meters (Nm). For now, we assme F is constant. We ll learn how to handle ariable F later. (Line integrals) (6) Applications: work: example Example 5: A 380 kg piano is pshed off the top of a bilding. It lands on the grond 40m below and 10m away from the bilding horizontally. Calclate the work done by graity as the piano falls. d F d = 10, 40 Magnitde of F is 9.8 m/s 380 kg = 374 kg m/s F points directly down: F = 0, 374. So the work done is W = F d = 10, 40 0, 374 = 148960 Nm
(7) Examples Example 6: Which of the following expressions are alid? (legal operations) w ( w) + ( w) No No No ( + ) w Yes (8) Examples Example 7: Which pairs from the following list of ectors are orthogonal? Which are parallel? = 1, 3, = 1, 0, w = 5, 1, 1 p = 0, 3, 0 q = 10,, Notice that q = w, so q and w are parallel. w = 0, so is orthogonal to both w and q. and p are orthogonal, and they are not orthogonal or parallel to any of, w, q.
(9) Examples Example 8: Find the angle θ between these two 5D ectors: = 0, 1, 3, 4, 5 = 1, 1, 1, 3, 6 Remember, yo can se the angle formla for the dot prodct: = cos θ. = 0( 1) + 1( 1) + ( 3)1 + ( 4)3 + 5 6 = 14 = 0 + 1 + ( 3) + ( 4) + 5 = 51 = ( 1) + ( 1) + 1 + 3 + 6 = 48 ( ) ( ) 14 θ = arccos = arccos 1.8 51 48 (30) Examples Example 9: Let a = a 1, a, a 3, and b = b 1, b, b 3 be distinct ectors in R 3. Show that the set of points (x, y, z) satisfying the following eqation is a sphere, and find its center and radis. ( x, y, z a) ( x, y, z b) = 0 0 = ( x, y, z a) ( x, y, z b) = x a 1, y a, z a 3 x b 1, y b, z b 3 = (x a 1 )(x b 1 ) + (y a )(y b ) + (z a 3 )(z b 3 ) = [ x (a 1 + b 1 )x + a 1 b 1 ] + [ y (a + b )y + a b ] + [ z (a 3 + b 3 )z + a 3 b 3 ] Complete the sqares: [ 0 = x a ] 1 + b ( 1 a1 b 1 [ + z a ] 3 + b ( 3 a3 b 3 ) [ + y a + b ) ] ( a b )
(31) Examples Example 9: (cont.) [ x a ] 1 + b [ 1 + y a ] + b [ + z a ] 3 + b 3 ( ) a1 b ( ) 1 a b ( ) a3 b 3 = + +. So the center has position ector a1 + b 1, a + b, a 3 + b 3 = 1 (a + b) and the radis is (a1 ) b ( ) 1 a b ( a3 b 3 r = + + ) = 1 a b. (3) Smmary The dot prodct of = 1,, 3 and = 1,, 3 is Angle formla: = 1 1 + + 3 3. = cos θ, where θ is the angle between the ectors. The dot prodct measres how mch two ectors agree. is zero if and only if and are orthogonal. The ector projection proj is the piece of that lies parallel to The scalar projection comp is the signed magnitde of proj. Work is the dot prodct of net force and distance: W = F d.
(33) Homework Paper homework 3 de Thrsday at the beginning of class WebAssign homework (1.1, 1.) de Satrday night, 11:59 PM Eeryone shold be able to access the corse now; let me know ASAP if anything goes wrong.