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Chapter 4 (A)

Chapter 4 (A). The Vector Space R n (4.1 ~ 4.3) Gareth Williams J & B, 滄海書局. Part A. 4.1 Introduction to Vectors 4.2 Dot Product, Norm, Angle, and Distance. the origin ： (0, 0) the position vector ： the initial point ： O the terminal point ： A (5, 3).

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Chapter 4 (A)

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  1. Chapter 4 (A) The Vector Space Rn (4.1 ~ 4.3) Gareth Williams J & B, 滄海書局

  2. Part A • 4.1 Introduction to Vectors • 4.2 Dot Product, Norm, Angle, and Distance

  3. the origin：(0, 0) • the position vector： • the initial point：O • the terminal point：A(5, 3) 4.1 Introduction to Vectors Rectangular Coordinate System Figure 4.1

  4. Example 1 Sketch the position vectors See Figure 4.2. Figure 4.2

  5. Figure 4.3

  6. Definition Let be a sequence of n real numbers. The set of all such sequences is called n-space and is called Rn. u1 is the first component of , u2 is the second component and so on.

  7. R4 is the collection of all sets of four ordered real numbers. For example, (1, 2, 3, 4) and are element of R4. R5 is the collection of all sets of five ordered real numbers. For example, is in this collection. Example 2

  8. Definition Let be two elements of Rn. We say that u and v are equal if u1 = v1, …, un = vn. Thus two element of Rn are equal if there corresponding components are equal.

  9. Definition Let be elements of Rn and let c be a scalar Addition and scalar multiplication are performed as follows Addition: Scalar multiplication :

  10. Example 3 Let u = ( –1, 4, 3, 7) and v = ( –2, –3, 1, 0) be vector in R4. Find u + v and 3u. Solution

  11. Example 4 Consider the vector (4, 1) and (2, 3), we get (4, 1) + (2, 3) = (6, 4). Figure 4.4

  12. In general, if u and v are vectors in the same vector space, then u+ v is the diagonal of the parallelogram defined by u and v. See Figure 4.5. Figure 4.5

  13. Example 5 Consider the scalar (3, 2) by 2, we get 2(3, 2) = (6, 4) Observe in Figure 4.6 that (6, 4) is a vector in the same direction as (3, 2), and 2 times it in the length. Figure 4.6

  14. c > 1 0 < c < 1 –1 < c < 0 c < –1 Figure 4.7

  15. Negative Vector The vector (–1)u is writing –u and is called the negative of u. Subtraction Subtraction is performed on element of Rn by subtracting corresponding components. Zero Vector The vector (0, 0, …, 0), having n zero components, is called the zero vector of Rn and is denoted 0.

  16. Figure 4.8 Commutativity of vector addition u + v = v + u Theorem 4.1 • Let u, v, and w be vectors in Rn and let c and d be scalars. • u + v = v + u • u + (v + w) = (u + v) + w • u + 0 = 0 + u = u • u + (–u) = 0 • c(u + v) = cu + cv • (c + d)u = cu + du • c(du) = (cd)u • 1u = u

  17. Theorem 4.1 Proof

  18. Example 6 Let u = (2, 5, –3), v = ( –4, 0, 2). Determine the vector 2u – 3v + w. Solution

  19. Figure 4.10 Example 7 Consider the vector (5, 4) and (7, 1), we get (5, 4) + (7, 1) = (12, 5). (5, 4) + (19, 6) = (24, 10). Figure 4.9

  20. Column Vectors We defined addition and scalar multiplication of column vectors in Rn in a componentwise manner: and

  21. 4.2 Dot Product, Norm, Angle, and Distance Definition Let be two vectors in Rn. The dot product of u and v is denoted u‧v and is defined by The dot product assigns a real number to each pair of vectors.

  22. Example 1 Find the dot product of u = (1, –2, 4) and v = (3, 0, 2) Solution

  23. Properties of the Dot Product • Let u, v, and w be vectors in Rn and let c be a scalar. Then • u‧v = v‧u • (u + v)‧w = u‧w + v‧w • cu‧v = c(u‧v) = u‧cv • u‧u 0, and u‧u = 0 if and only if u = 0

  24. Properties of the Dot Product Proof

  25. Norm of a Vector in Rn Definition The norm (length or magnitude) of a vector u = (u1, …, un) in Rn is denoted ||u|| and defined by Note: The norm of a vector can also be written in terms of the dot product Figure 4.11

  26. Example 2 Find the norm of the vectors u = (1, 3, 5) of R3 and v = (3, 0, 1, 4) of R4. Solution

  27. Definition A unit vector is a vector whose norm is 1. If v is a nonzero vector, then the vector is a unit vector in the direction of v. This procedure of constructing a unit vector in the same direction as a given vector is called normalizing the vector.

  28. Example 3 • Show that the vector (1, 0) is a unit vector. • Find the norm of the vector (2, –1, 3). Normalize this vector. Solution

  29. Unit Vectors • (1, 0), (0, 1) are unit vectors in R2. • (1, 0, 0 ), (0, 1, 0) , (0, 0, 1) are unit vectors in R3. • (1, 0 , 0 , 0 ), (0, 1, 0 , 0) , (0, 0, 1 , 0) , (0, 0, 0 , 1) are unit vectors in R4. • (1, 0, … , 0 ), … …, (0, … , 0 , 1) are unit vectors in Rn.

  30. Theorem 4.2 The Cauchy-Schwartz Inequality. If u and v are vectors in Rn then Here denoted the absolute value of the number

  31. The Cauchy-Schwartz Inequality Proof

  32. Figure 4.12 Angle between Vectors

  33. Definition Let u and v be two nonzero vectors in Rn. The cosine of the angle  between there vector is

  34. Example 4 Determine the angle between the vectors u = (1, 0, 0) and v = (1, 0, 1) in R3. Solution

  35. Theorem 4.3 The nonzero vectors u and v are orthogonal if and only if u‧v = 0.

  36. Example 5 • Show that the following pairs of vectors are orthogonal. • (1, 0) and (0, 1). • (2, –3, 1) and (1, 2, 4). Solution

  37. Pairwise Orthogonal Vectors • (1, 0), (0, 1) are orthogonal vectors in R2. • (1, 0, 0 ), (0, 1, 0) , (0, 0, 1) are orthogonal vectors in R3. • (1, 0, … , 0 ), … …, (0, … , 0 , 1) are orthogonal vectors in Rn.

  38. Figure 4.13 Example 6 Determine a vector in R2 that is orthogonal to (3, –1). Show that there are many such vectors and that they all lie on a line. Solution

  39. Example 7 (a) Show that following vectors are orthogonal. (b) Compute the norm of each vector. Solution

  40. Figure 4.14(a) Figure 4.14(b) Theorem 4.4 • Let u and v be vectors in Rn. • Triangle Inequality: • This inequality tells us that the length of one side of a triangle cannot exceed the sum of the lengths of the order two sides. • Pythagorean theorem: the u‧v = 0 then ||u + v||2 = ||u||2 + ||v||2. The square of the hypotenuse of a right triangle is equal to the sum if the squares of the other two sides.

  41. Triangle Inequality Proof

  42. Pythagoreantheorem Proof

  43. Distance between Points Let be two points in Rn. The distance between x and y is denoted d(x, y) and is defined by Note: We can also write this distance as follows.

  44. Example 8 Determine the distance between the points x = (1,–2 , 3, 0) and y = (4, 0, –3, 5) in R4. Solution

  45. Figure 4.15

  46. Example 9 Prove that the distance function on Rn has the following symmetric property:d(x, y) = d(y, x). Solution

  47. Euclidean Geometry of Rn • Dot product of vectors u and v: u．v =u1v1+…+unvn • Norm of a vector u: • Angle between vectors u and v: • Distance between points x and y:

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