Math 221: LINEAR ALGEBRA §4-1. Vectors and Lines Le Chen 1 Emory University, 2020 Fall (last updated on 08/18/2020) Creative Commons License (CC BY-NC-SA) 1 Slides are adapted from those by Karen Seyffarth from University of Calgary.
A vector quantity has both magnitude and direction; e.g. displacement, force, wind velocity. Whereas two scalar quantities are equal if they are represented by the same value, two vector quantities are equal if and only if they have the same magnitude and direction. Scalar quantities versus vector quantities ◮ A scalar quantity has only magnitude; e.g. time, temperature.
force, wind velocity. Whereas two scalar quantities are equal if they are represented by the same value, two vector quantities are equal if and only if they have the same magnitude and direction. Scalar quantities versus vector quantities ◮ A scalar quantity has only magnitude; e.g. time, temperature. ◮ A vector quantity has both magnitude and direction; e.g. displacement,
force, wind velocity. Whereas two scalar quantities are equal if they are represented by the same value, two vector quantities are equal if and only if they have the same magnitude and direction. Scalar quantities versus vector quantities ◮ A scalar quantity has only magnitude; e.g. time, temperature. ◮ A vector quantity has both magnitude and direction; e.g. displacement,
space, respectively. R 2 and R 3 Vectors in R 2 and R 3 have convenient geometric representations as position vectors of points in the 2-dimensional (Cartesian) plane and in 3-dimensional
The vector . The vector z R 3 c R 2 ( a , b , c ) y ( a , b ) b 0 b x 0 a y a x � � a b a b . c
If is a point in , then is often used to denote the position vector of the point. Instead of using a capital letter to denote the vector (as we generally do with matrices), we emphasize the importance of the geometry and the direction with an arrow over the name of the vector. Notation ◮ If P is a point in R 3 with coordinates ( x , y , x ) we denote this by P = ( x , y , z ) .
direction with an arrow over the name of the vector. is often used to denote the position vector of the point. with matrices), we emphasize the importance of the geometry and the Instead of using a capital letter to denote the vector (as we generally do Notation ◮ If P is a point in R 3 with coordinates ( x , y , x ) we denote this by P = ( x , y , z ) . ◮ If P = ( x , y , z ) is a point in R 3 , then x − → 0 P = y z
direction with an arrow over the name of the vector. is often used to denote the position vector of the point. with matrices), we emphasize the importance of the geometry and the Notation ◮ If P is a point in R 3 with coordinates ( x , y , x ) we denote this by P = ( x , y , z ) . ◮ If P = ( x , y , z ) is a point in R 3 , then x − → 0 P = y z ◮ Instead of using a capital letter to denote the vector (as we generally do
is associated with the point . in and , and we say that both and the point Often, there is no distinction made between the vector . Any vector Notation and Terminology ◮ The notation − → 0 P emphasizes that this vector goes from the origin 0 to the point P . We can also use lower case letters for names of vectors. In this case, we write − → 0 P = � p .
. Often, there is no distinction made between the vector and , and we say that both and the point Notation and Terminology ◮ The notation − → 0 P emphasizes that this vector goes from the origin 0 to the point P . We can also use lower case letters for names of vectors. In this case, we write − → 0 P = � p . ◮ Any vector x 1 in R 3 � x = x 2 x 3 is associated with the point ( x 1 , x 2 , x 3 ) .
. and we say that both and Notation and Terminology ◮ The notation − → 0 P emphasizes that this vector goes from the origin 0 to the point P . We can also use lower case letters for names of vectors. In this case, we write − → 0 P = � p . ◮ Any vector x 1 in R 3 � x = x 2 x 3 is associated with the point ( x 1 , x 2 , x 3 ) . ◮ Often, there is no distinction made between the vector � x and the point ( x 1 , x 2 , x 3 ) ,
Notation and Terminology ◮ The notation − → 0 P emphasizes that this vector goes from the origin 0 to the point P . We can also use lower case letters for names of vectors. In this case, we write − → 0 P = � p . ◮ Any vector x 1 in R 3 � x = x 2 x 3 is associated with the point ( x 1 , x 2 , x 3 ) . ◮ Often, there is no distinction made between the vector � x and the point x 1 ∈ R 3 . ( x 1 , x 2 , x 3 ) , and we say that both ( x 1 , x 2 , x 3 ) ∈ R 3 and � x = x 2 x 3
. Analogous results hold for In this case, , i.e., Theorem x x 1 and � be vectors in R 3 . Then Let � v = y w = y 1 z z 1
. Analogous results hold for In this case, , i.e., Theorem x x 1 and � be vectors in R 3 . Then Let � v = y w = y 1 z z 1 v = � w if and only if x = x 1 , y = y 1 , and z = z 1 . 1. �
. Analogous results hold for In this case, , i.e., Theorem x x 1 and � be vectors in R 3 . Then Let � v = y w = y 1 z z 1 v = � w if and only if x = x 1 , y = y 1 , and z = z 1 . 1. � x 2 + y 2 + z 2 . � 2. || � v || =
. Analogous results hold for In this case, , i.e., Theorem x x 1 and � be vectors in R 3 . Then Let � v = y w = y 1 z z 1 v = � w if and only if x = x 1 , y = y 1 , and z = z 1 . 1. � x 2 + y 2 + z 2 . � 2. || � v || = v = � 3. � 0 if and only if || � v || = 0 .
. Analogous results hold for In this case, , i.e., Theorem x x 1 and � be vectors in R 3 . Then Let � v = y w = y 1 z z 1 v = � w if and only if x = x 1 , y = y 1 , and z = z 1 . 1. � x 2 + y 2 + z 2 . � 2. || � v || = v = � 3. � 0 if and only if || � v || = 0 . 4. For any scalar a, || a � v || = | a | · || � v || .
Theorem x x 1 and � be vectors in R 3 . Then Let � v = y w = y 1 z z 1 v = � w if and only if x = x 1 , y = y 1 , and z = z 1 . 1. � x 2 + y 2 + z 2 . � 2. || � v || = v = � 3. � 0 if and only if || � v || = 0 . 4. For any scalar a, || a � v || = | a | · || � v || . w ∈ R 2 , i.e., Analogous results hold for � v , � � x � x 1 � � � v = , � w = . y y 1 x 2 + y 2 . � In this case, || � v || =
Example � − 3 3 − 6 � Let � p = , � q = − 1 , and − 2 � q = 2 , 4 − 2 4
Example � − 3 3 − 6 � Let � p = , � q = − 1 , and − 2 � q = 2 , 4 − 2 4 Then √ � ( − 3) 2 + 4 2 = || � p || = 9 + 16 = 5 ,
Example � − 3 3 − 6 � Let � p = , � q = − 1 , and − 2 � q = 2 , 4 − 2 4 Then √ � ( − 3) 2 + 4 2 = || � p || = 9 + 16 = 5 , √ √ (3) 2 + ( − 1) 2 + ( − 2) 2 = � || � q || = 9 + 1 + 3 = 14 ,
Example � − 3 3 − 6 � Let � p = , � q = − 1 , and − 2 � q = 2 , 4 − 2 4 Then √ � ( − 3) 2 + 4 2 = || � p || = 9 + 16 = 5 , √ √ (3) 2 + ( − 1) 2 + ( − 2) 2 = � || � q || = 9 + 1 + 3 = 14 , and ( − 6) 2 + 2 2 + 4 2 � || − 2 � q || = √ = 36 + 4 + 16 √ √ = 56 = 4 × 14 √ = 2 14 = 2 || � q || .
Geometric Vectors Let A and B be two points in R 2 or R 3 . y • − → B AB is the geometric vector from A to B . • A is the tail of − → AB . • B is the tip of − → AB . • the magnitude of − → AB is its length, and is denoted ||− → AB || . 0 x A
y B • − → AB is the vector from A (1 , 0) to B (2 , 2) . • − → CD is the vector from C ( − 1 , − 1) D to D (0 , 1) . 0 x A • − AB = − → → CD because the vectors have C the same length and direction.
We co-ordinatize vectors by putting them in standard position, and then identifying them with their tips. Definition A vector is in standard position if its tail is at the origin. y P B 0 x A � 1 � Thus − AB = − → 0 P where P = P (1 , 2) , and we write − → → = − → 0 P = AB . 2 − → 0 P is the position vector for P (1 , 2) .
If we aren’t concerned with the locations of the tail and tip, we simply write x More generally, if P ( x , y , z ) is a point in R 3 , then − → 0 P = y is the position z vector for P .
If we aren’t concerned with the locations of the tail and tip, we simply write x More generally, if P ( x , y , z ) is a point in R 3 , then − → 0 P = y is the position z vector for P . x � v = y z
parallelogram law direction opposite to . and and having the same tail as , and is the diagonal of the parallelogram defjned by addition: . if ; vector equality: same length and direction. if the same direction as and has length , then , and scalar multiplication: if : the vector with length zero and no direction. Intrinsic Description of Vectors
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