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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.

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Geometry of R 2 and R 3

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  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

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