Class Note for MATH 215 at UA
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Date Created: 02/06/15
THE UNIVERSITY Q OF ARIZONA Math 215 Introduction to Linear Algebra Section 12 Length amp Angle The Dot Product Paul Dostert August 15 2008 A Definition of the Dot Product If u U17 LL27 7um and v 1117 v2 7 11 then the dotproduct u v Of u and v is defined by u39VUlv1 U2U2Unvn This is the sum of products of each component of the vector Note a u and v must have the same size b u v produces a number For this reason it is often called the scalar product c The dot product is an example of an innerproduct a concept very important to mathematical analysis Ex Compute u v for a u 17 amp V 0737 b u 5 37 2 ampv 3717 170 At Properties of the Dot Product Thm Let u7v amp w be vectors in R and c a scalar Then a uvvu b uvWuvuW C cuVCuv d uuZOanduu0iffu0 The length or norm of a vector v v17 7111 in R is vx vv vvg Thm Let v be a vector in R and let 0 be a scalar Then a 0 iffv 0 b IICVII CV Ex Compute 171727 3 At Unit Vectors A vector of length 1 is called a unit vector In order to create a unit vector we divide any given vector by its length u 1v v This process is called normalizing the vector The vectors e17 e27 7 en e R where ez is zero except for a 1 in the ith component are called the standard unit vectors or standard basis vectors Note in R2 and R3 these correspond to the axes Thm For all vectors u7v e Rquot u v g CauchySchwarz Inequality and u v g The Triangle Inequality Note These are very important inequalities to remember when attempting to prove many properties of vectors especially error estimates A Distance and Angle The distance d u7 v between vectors u7v e R is given by dltu7vgt llu vll Wm v1gt2ltun vngt2 Note Since these are vectors in standard position this is just the formula for the distance between two points the head points in Rquot The angle between u7 v e R is given by C086 i uV Note I will expect you to know 9 corresponding to cos 7r2 0 cosO 1 and cos7r 1 ONLY Use your calculator to find other angles Angles should be in radians unless requested otherwise Ex Compute the angle between each pair of vectors a 27 07 2 7 07 27 2 b 17070717070 C 17 170717170 At Orthogonality and Projections Two vectors u7 v e R are orthogonal to each other if u v 0 Pythagoras Thm For all uv e Rquot uv2 u2 v2 iff u and v are orthogonal If u7 v e R and u 7 0 then the projection ofv onto u is proju V W11 This represents the portion of the vector v that is in the direction of u Note 0 I prefer to write the projection as u llull2 proju v u v since this is clearly a scalar u v times a unit vector in the direction of u o If the angle between u and v is obtuse then the projection will be in the opposite direction of u A Examples Ex Find the projection of v onto u for a u 27 1 7v 173 b u270727v17171 C u17372ue117070 Ex Prove d u7 v 0 iff u v Ex Let A 1717 1 B 37 27 2 and C39 27 27 4 Prove that AABC39 is a right angled triangle Ex Find scalar k so that u 17 17 2 and v k27 k7 3 are orthogonal A Matlab Examples Ex For u 1171 amp v 3 2 0 compute u v and projuv The dot product between two vectors is given by the function dot It takes the two vectors as arguments To find u we use the norm function u111 v3 20 dotuv dotuvunormu Ex Find the angle between u 1 3 5 47 49 and v 2524321 There are a few tricks in this one First use the shortcut way to form u as discussed in the previous lecture For v we use something similar We write 11 25 1 1 which means go from 25 by 1 to 1 We then have cosQ V To solve for 9 we use the arccosine function which is acos in llullllvll Matlab u1249 v25 11 theta acos dotuv normu normv
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