1 / 30

Geometry of R 2 and R 3

Geometry of R 2 and R 3. Vectors in R 2 and R 3. NOTATION. R The set of real numbers R 2 The set of ordered pairs of real numbers R 3 The set of ordered triples of real numbers. Vector.

onslow
Download Presentation

Geometry of R 2 and R 3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Geometry of R2 and R3 Vectors in R2 and R3

  2. NOTATION • R The set of real numbers • R2The set of ordered pairs of real numbers • R3The set of ordered triples of real numbers

  3. Vector • A vector in R2 (or R3) is a directed line segment from the origin to any point in R2 (or R3) • Vectors in R2 are represented using ordered pairs • Vectors in R3 are represented using ordered triples

  4. Notation for Vectors • Vectors in R2 (or R3) are denoted using bold faced, lower case, English letters • Vectors in R2 (or R3) are written with an arrow above lower case, English letters • Points in R2 (or R3) are denoted using upper case English letters

  5. Example 1 • u = (u1, u2, u3) represent a vector in R3from the origin to the point P (u1, u2, u3) • u1, u2, and u3 are the components of the u

  6. Equality of Two Vectors • Two vectors are equal if their corresponding components are equal. • That is, u = (u1, u2, u3) and v = (v1, v2, v3) are equal if and only if u1 = v1, u2 = v2, and u3 = v3 • Hence, if u = 0, the zero vector, then u1 = u2 = u3 = 0.

  7. Collinear Vectors • Two vectors are collinear if thy both lie on the same line. • That is, u = (u1, u2, u3) and v = (v1, v2, v3) are collinear if the points U, V, and the Origin are collinear points.

  8. Length of a Vector in R2 The length (norm, magnitude) of v = (v1, v2), denoted by ||v||, is the distance of the point V (v1, v2) from the origin.

  9. Length of a Vector in R3 The length (norm, magnitude) of v = (v1, v2, v3) is the distance of the point V (v1, v2, v3) from the origin.

  10. Example Find the length of u = (-4, 3, -7)

  11. Zero Vector and Unit Vector • The magnitude of 0 is zero. • If a vector has length zero, then it is 0 • If a vector has magnitude 1, it is called a unit vector.

  12. Scalar Multiplication Let c be a scalar and u a vector in R2 (or R3). Then the scalar multiple of u by c is the vector the vector obtained by multiplying each component of u by c. That is, cu = (cu1, cu2) in R2, and cu = (cu1, cu2, cu3) in R3

  13. Example Find cu for u = (-4, 0, 5) and c = 2. If v = (-1, 1), sketch v, 2v and -2v.

  14. Theorem 1.1.1 Let u be a nonzero vector in R2 or R3, and c be any scalar. Then u and cu are collinear, and • if c > 0, then u and cu have the same direction • if c < 0, then u and cu have opposite directions • ||cu|| = |c| ||u||

  15. Example Let u = (-4, 8, -6) • Find the midpoint of the vector u. • Find a the unit vector in the direction of u. • Find a vector in the direction opposite to u that is 1.5 times the length of u.

  16. Vector Addition Let u and v be nonzero vectors in R2 or R3.Then the sum u + v is obtained by adding the corresponding components. That is, • u + v = (u1 + v1, u2 + v2), in R2 • u + v = (u1 + v1, u2 + v2, u3 + v3), in R3

  17. Example Find the sum of each pair of vectors • u = (2, 1, 0) and v = (-1, 3, 4) • u = (1, -2) and v = (-2, 3) Sketch each vector in part (2) and their sum.

  18. Theorem 1.1.2 For nonzero vectors u and v the directed line segment from the end point of u to the endpoint of u + v is parallel and equal in length of v.

  19. Proof of Theorem 2: Outline • Show that d(u, u+v) = d(0, v). • Show that d(v, u+v) = d(0, u). • The above two parts proves that the four line segments form a parallelogram. • The opposite sides of a parallelogram are parallel and of the same length. (A result from Geometry.) • We must also prove that the four vectors u, v, u + v, and 0 are coplanar, which will be done in section 1.2.

  20. Opposite and Vector Subtraction Let u be vector in R2 or R3. Then • Opposite or Negative of u, denoted by –u, is (-1)(u). • The difference u – v is defined as u +(–v).

  21. Theorem 1.1.3 Let u, v and w be vectors in R2 or R3, and c and d scalars. Then • u + v = v + u • (u + v) + w = v + (u + w) • u + 0 = u • u + (-u) = 0 • (cd)u = c(du)

  22. Theorem 3 Cont’d. Let u, v and w be vectors in R2 or R3, and c and d scalars. Then • (c + d)u = cu + du • c(u + v) = cu + cv • 1u = u • (-1)u = -u • 0u = 0

  23. Equivalent Directed Line Segments Two directed line segments are said to be equivalent if they have the same direction and length.

  24. Theorem 1.1.4 Let U and V be distinct points in R2 or R3. Then the vector v – u is equivalent to the directed line segment from U to V. That is, • The line UV is parallel to the vector v – u, and • d(u, v) = ||v – u||

  25. Proof of Theorem 4: Outline • Show that the sum of u and v – u is v. • This proves that the two vectors v – u is parallel and equal in length to the directed line segment from U to V.

  26. Example Is the line determined by (3,1,2) & (4,3,1), parallel to the line determined by (1,3,-3) & (-1,-1,-1)? Outline for the solution: Find unit vectors in the direction of the lines. If they are same or opposite, then the two vectors are parallel.

  27. Standard Basis Vectors in R2 i = (1, 0) j = (0, 1) If (a, b) is a vector in R2, then (a, b) = a(1, 0) + b(0, 1) = ai + bj.

  28. Standard Basis Vectors in R3 i = (1, 0, 0) j = (0, 1, 0) k = (0, 0, 1) If (a, b, c) is a vector in R3, then (a, b, c) = ai + bj + ck

  29. Example Express (2, 0, -3) in i, j, k form.

  30. Homework 1.1

More Related