Vectors in Three-Dimensional Space Vectors By a vector a,we mean an ordered triple of real numbers: a=(a1,a2,a3) The numbers a,a2,a3 are called the components ofa. 2 P(a,a2,a3) 0 Two vectors are "equal"if they have the same components; (a,a2,a3)=(b,b2,bg)iffa1=b1,a2=b2,a3=b3, Main Meny o墙8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Vectors in Three-Dimensional Space Vectors By a vector a, we mean an ordered triple of real numbers: a = (a1 , a2 , a3 ). The numbers a1 , a2 , a3 are called the components of a. Two vectors are “equal” if they have the same components; (a1 , a2 , a3 ) = (b1 , b2 , b3 ) iff a1 = b1 , a2 = b2 , a3 = b3
for a=(a1,a2,a3）) and b=(b1,b2,b3) (13.2.1) we define a+b=(a1+b1,a2+b2,a3+b3). 个 a+b=(a,+b,a2+b2,a3+b) b=(b,b2,b3) d=(a,a2,a3) Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. ( , , ) a = a 1 a 2 a 3 ( , , ) b = b1 b2 b3 ( , , ) a + b = a 1 + b1 a 2 + b2 a 3 + b3
for a=(a1,a2,a3) and o real (13.2.2) we define aa=(aa,ad2,aa3). d=(aa1,0a2,aa3） a=(a,a2,a3) Main Meny 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. ( , , ) a = a 1 a 2 a 3 ( , , ) a = a 1 a 2 a 3
Vectors in Three-Dimensional Space Visualizinga+b For a=(a1,a2,a3)and b=(b.b2,b3)we defined a+b=(a1+b,a2+b,a3+b3). a+b a+b Figure 13.2.2 Figure13.2.3 Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Vectors in Three-Dimensional Space Visualizing a + b For a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ) we defined a + b = (a1 + b1 , a2 + b2 , a3 + b3 )
Vectors in Three-Dimensional Space Norm By the norm (magnitude,length)of the vector a=(a,az,a3)we mean the number (13.2.3) lal=√a+a+a, Figure 13.2.4 (1)lal≥0 and llall =0 iff a =0. (13.2.4) (2)lloall l1 llall. (3)la+bl≤la‖+b (the triangle inequality) Main Meny 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Vectors in Three-Dimensional Space Norm By the norm (magnitude, length) of the vector a = (a1 , a2 , a3 ) we mean the number
Vectors in Three-Dimensional Space For a0,the vector b=aa is the unique vector of length (13.2.5) lalall,which has the direction of a ifa>0 and the opposite direction ifa<0. Figure 13.2.5 Main Meny p,5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Vectors in Three-Dimensional Space
Vectors in Three-Dimensional Space b b a -b Figure 13.2.6 a+6 Figure 13.2.7 Figure 13.2.8 Main Meny o墙8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Vectors in Three-Dimensional Space
Vectors in Three-Dimensional Space Parallel Vectors For a0,the vectors parallel to a are by definition (13.2.6) the scalar multiples aa. (13.2.7) By special convention,all vectors are said to be parallel to 0. if a and b are both parallel to c,then every linear combination (13.2.8) aa+Bb is also parallel to c. Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Vectors in Three-Dimensional Space Parallel Vectors
Vectors in Three-Dimensional Space -2 -1 a=(2,-2,6) b=(1,-1,3) c=(-1,1,-3) some parallel vectors Figure 13.2.9 Main Menu 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Vectors in Three-Dimensional Space
Vectors in Three-Dimensional Space Unit Vectors Vectors of norm 1 are called unit vectors. For each nonzero vector a there is a unique unit vectoru which has the direction of a. 1 a (13.2.9) Ua= Da*=a Consider the vectors i=(1,0,0),j=(0,1,0,k=(0,0,1). Main Meny C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Vectors in Three-Dimensional Space Unit Vectors Vectors of norm 1 are called unit vectors. For each nonzero vector a there is a unique unit vector ua which has the direction of a. Consider the vectors i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1)