A FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION

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1 A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler R n : Vectors Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION Attribution-NonCommercial-ShareAlike (CC BY-NC-SA) This license lets others remix, tweak, and build upon your work non-commercially, as long as they credit you and license their new creations under the identical terms. R n : Vectors Page 1/31

2 What is R n? Notation and Terminology R denotes the set of real numbers. R denotes the set of all column vectors with two entries. R 3 denotes the set of all column vectors with three entries. In general, R n denotes the set of all column vectors with n entries. R n : Vectors What is Rn? Page /31

3 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. R n : Vectors What is Rn? Page 3/31

4 R and R 3 Vectors in R and R 3 have convenient geometric representations as position vectors of points in the -dimensional (Cartesian) plane and in 3-dimensional space, respectively. R n : Vectors What is Rn? Page 4/31

5 z R y R 3 c (a, b, c) b (a, b) 0 a x 0 b y a [ a The vector b ]. x The vector a b c. R n : Vectors What is Rn? Page 5/31

6 Notation If P is a point in R n with coordinates (p 1, p,..., p n ) we denote this by P = (p 1, p,..., p n ). If P = (p 1, p,..., p n ) is a point in R n, then p 1 p 0P =. p n 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. R n : Vectors What is Rn? Page 6/31

7 Notation and Terminology The notation 0P 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 0P = p. Any vector x = x 1 x. x n is associated with the point (x 1, x,..., x n ). in Rn Often, there is no distinction made between the vector x and the point (x 1, x,..., x n ), and we say that both (x 1, x,..., x n ) R n and x 1 x x =. Rn. x n R n : Vectors What is Rn? Page 7/31

8 Geometric Vectors in R and R 3 Let A and B be two points in R or R 3. y B 0 x 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. A R n : Vectors Geometric Vectors Page 8/31

9 Equality of geometric vectors y D B AB is the vector from A = (1, 0). to B = (, ). CD is the vector from C = ( 1, 1) to D = (0, 1). C 0 A x AB = CD because the vectors have the same length and direction. The fact that the points A and B are different from the points C and D is not important. For geometric vectors, the location of the vector in the plane (or in 3-dimensional space) is not important; the important properties are its length and direction. R n : Vectors Geometric Vectors Page 9/31

10 Coordinatizing Vectors Part 1 y P B 0P is the position vector for P = (1, ), and [ ] 1 0P =. 0 A x Since AB = 0P, it should be the case that [ 1 AB = moving AB so that its tail is at the origin. ]. This can be seen by A geometric vector is coordinatized by putting it in standard position, meaning with its tail at the origin, and then identifying the vector with its tip. R n : Vectors Geometric Vectors Page 10/31

11 Algebra in R n Addition in R n Since vectors in R n are n 1 matrices, addition in R n is precisely matrix addition using column matrices, i.e., If u and v are in R n, then u + v is obtained by adding together corresponding entries of the vectors. Example Let u = The zero vector in R n is the n 1 zero matrix, and is denoted and v = u + v =. Then, = R n : Vectors Algebra in Rn Page 11/31

12 Properties of Vector Addition Let u, v, and w be vectors in R n. Then the following properties hold. 1 u + v = v + u (vector addition is commutative). ( u + v) + w = u + ( v + w) (vector addition is associative). 3 u + 0 = u (existence of an additive identity). 4 u + ( u) = 0 (existence of an additive inverse). R n : Vectors Algebra in Rn Page 1/31

13 Scalar Multiplication Since vectors in R n are n 1 matrices, scalar multiplication in R n is precisely matrix scalar multiplication using column matrices, i.e., If u is a vector in R n and k R is a scalar, then k u is obtained by multiplying every entry of u by k. Example Let u = 1 3 and k = 4. Then, k u = = R n : Vectors Algebra in Rn Page 13/31

14 Properties of Scalar Multiplication Let u, v R n be vectors and k, p R be scalars. Then the following properties hold. 1 k( u + v) = k u + k v (scalar multiplication distributes over vector addition). (k + p) u = k u + p u (addition distributes over scalar multiplication). 3 k(p u) = (kp) u (scalar multiplication is associative). 4 1 u = u (existence of a multiplicative identity). R n : Vectors Algebra in Rn Page 14/31

15 The Geometry of Vector Addition 1 Vector Equality. The vectors have the same length and direction. The zero vector, 0 has length zero and no direction. 3 Addition. Let u, v be vectors. Then u + v is the diagonal of the parallelogram defined by u and v, and having the same tail as u and v. u u + v v R n : Vectors The Geometry of Vector Addition Page 15/31

16 Tip-to-Tail Method for Vector Addition For points A, B and C, AB + BC = AC. AB C AC BC BC A AB B R n : Vectors The Geometry of Vector Addition Page 16/31

17 Example The diagonals of any parallelogram bisect each other. To see this, denote the parallelogram by its vertices, ABCD. A B M D C Let M denote the midpoint of AC. Then AM = MC. It now suffices to show that BM = MD. BM = BA + AM = CD + MC = MC + CD = MD. Since BM = MD, these vectors have the same magnitude and direction, implying that M is the midpoint of BD. Therefore, the diagonals of ABCD bisect each other. R n : Vectors The Geometry of Vector Addition Page 17/31

18 The Geometry of Vector Subtraction Let u and v be vectors in R or R 3. The vector u v = u + ( v) is obtained from the parallelogram defined by u and v by taking the vector from the tip of v to the tip of u, i.e., the diagonal of the parallelogram, directed towards the tip of u. u v u v v u R n : Vectors The Geometry of Vector Addition Page 18/31

19 Coordinatizing Vectors Part Let A = (x 1, y 1, z 1 ) and B = (x, y, z ) be two points in R 3. B z x We see from the figure that 0A + AB = 0B, and hence AB = 0B 0A = 0 x y z A x 1 y 1 z 1 y = x x 1 y y 1 z z 1. R n : Vectors The Geometry of Vector Addition Page 19/31

20 Length of a Vector The Distance Between Points For A = (x 1, y 1, z 1 ) and B = (x, y, z ) in R 3, the distance between them is written d(a, B) and is given by d(a, B) = (x x 1 ) + (y y 1 ) + (z z 1 ). This is called the distance formula If P = (x x 1, y y 1, z z1), then the distance between the origin and P is equal to the the distance between points A and B i.e., d(0, P) = d(a, B) = (x x 1 ) + (y y 1 ) + (z z 1 ). R n : Vectors Length of a Vector Page 0/31

21 Properties of Distance Let P and Q be two points in R n, and d(p, Q) the distance between them. Then the following properties hold. 1 The distance between P and Q is equal to the distance between Q and P, i.e., d(p, Q) = d(q, P). d(p, Q) 0 with equality if and only if P = Q. Example For P = (1, 1, 3) and Q = (3, 1, 0), the distance between P and Q is d(p, Q) = + + ( 3) = 17. R n : Vectors Length of a Vector Page 1/31

22 Length of a Vector More generally, if P = (p 1, p,..., p n ) and Q = (q 1, q,..., q n ) are points in R n, then the distance between P and Q is the length of the vector PQ, written PQ. If x = d(p, Q) = PQ = (q 1 p 1 ) + (q p ) + + (q n p n ). ] R x, y X = (x 1, x ) x [ x1 0 x then the length of the vector x is the distance from the origin 0 to the point X = (x 1, x ) given by d(0, X ). The length of x, denoted x, is given by: d(0, X ) = x = x1 + x. R n : Vectors Length of a Vector Page /31

23 The formula for calculating the length of a vector generalizes to R n : if x 1 x x =. Rn, x n then x = x 1 + x + + x n, which represents the distance from the origin to the point (x 1, x,..., x n ). R n : Vectors Length of a Vector Page 3/31

24 Example [ ] 3 Let p = and q = The lengths of these vectors are. Then q = ( ) q = 6 4. p = ( 3) + 4 = = 5, and q = (3) + ( 1) + ( ) = = 14, q = ( 6) = = 56 = 4 14 = 14 = q. R n : Vectors Length of a Vector Page 4/31

25 Unit Vectors Definition A unit vector is a vector of length one. Example 1 0 0, Example If v 0, then, 0 0 1, 0, are examples of unit vectors. 1 v v is a unit vector in the same direction as v. R n : Vectors Length of a Vector Page 5/31

26 Example 1 v = 3 is not a unit vector, since v = 14. However, u = 1 v = 14 is a unit vector in the same direction as v, i.e., u = 1 14 v = = 1. Example If v and w are nonzero that have the same direction, then v = v w w; opposite directions, then v = v w w. R n : Vectors Length of a Vector Page 6/31

27 The Geometry of Scalar Multiplication Scalar Multiplication. If v 0 and a R, a 0, then a v has length a v = a v, and has the same direction as v if a > 0; has direction opposite to v if a < 0. Parallel Vectors. Two nonzero vectors are called parallel if they have the same direction or opposite directions. It follows that nonzero vectors v and w are parallel if and only if one is a scalar multiple of the other. R n : Vectors The Geometry of Scalar Multiplication Page 7/31

28 Problem Let P = (1,, 1),Q = ( 3, 0, 5), X = (, 1, 5) and Y = (4,, 3) be points in R 3. Is PQ parallel to XY? Is PX parallel to QY? Solution 4 PQ =, XY = 4 some scalar k, i.e., 4 4 1, and these vectors are parallel if PQ = k XY for = k 1 or 4 4 = k k k This gives a system of three equations in one variable, which is consistent, and has unique solution k =. Therefore, PQ is parallel to XY. PX =, QY =, and these vectors are parallel if PX = l QY for some scalar l. You will find that no such l exists, so PX is not parallel to QY.. R n : Vectors The Geometry of Scalar Multiplication Page 8/31

29 Vector problems and examples Problem Find the point, M, that is midway between P 1 = ( 1, 4, 3) and P = (5, 0, 3). Solution P 1 M P 0M = 0P1 + P 1 M = 0P1 + 1 P 1 P = 0 = = Therefore M = (,, 0). R n : Vectors Vector problems and examples Page 9/31

30 Problem Find the two points trisecting the segment between P = (, 3, 5) and Q = (8, 6, ). Solution P A B Q 0A = 0P + 1 PQ 3 0B = 0P + PQ 3 Since PQ = 0A = , 3 1 = and 0B = Therefore, the two points are A = (4, 0, 4) and B = (6, 3, 3). 4 6 = R n : Vectors Vector problems and examples Page 30/31

31 Example If ABCD is an arbitrary quadrilateral, then the the midpoints of the four sides of ABCD are the vertices of a parallelogram. A M 1 M 4 B D It suffices to prove that M 1 M = M 4 M 3. M M 3 C Let M 1 denote the midpoint of AB, M the midpoint of BC, M 3 the midpoint of CD, and M 4 the midpoint of DA. M 1 M = M 1 B + BM M 4 M 3 = M 4 D + DM 3 = 1 AB + 1 BC = 1 AD + 1 DC = 1 1 AC = AC Since M 1 M = M 4 M 3, the points M 1, M, M 3, M 4 are the vertices of a parallelogram. R n : Vectors Vector problems and examples Page 31/31