F24_Linear_Algebra_-_3.3_Orthogonality lOT

Linear Algebra - Section 3.3

1. Overview

This section covers:

• Orthogonal vectors

• Lines and planes

• Normal vectors

• Orthogonal projections

• Distance between a point and a plane

• Distance between parallel planes

2. Angles and Dot Product

• If two vectors u and v are perpendicular, then their angle θ = π/2 (90°).

• Relationship between angle and dot product:

  • If θ = π/2, then u · v = 0.

3. Definition: Orthogonal Vectors

• Two nonzero vectors u and v in Rⁿ are orthogonal if:

  • u · v = 0.

• The zero vector in Rⁿ is orthogonal to every vector in Rⁿ.

4. Examples of Orthogonal Vectors

Example #1

(a) Show that u = (−2, 3, 1, 4) and v = (1, 2, 0, −1) are orthogonal.- Calculation:[ u · v = (-2)(1) + (3)(2) + (1)(0) + (4)(-1) = 0 ](b) Show standard unit vectors i, j, k in R³ are orthogonal.- i · j = 0, j · k = 0, i · k = 0.

5. Lines and Planes

• Describing lines in R²:

  • Slope-Intercept Form: y = mx + b

  • Point Slope Form: y - y₀ = m(x - x₀)

  • Standard Form: ax + by = c

• Point-Normal Form of a Line:

  • Line containing point P₀(x₀, y₀) with normal vector n = (a, b) is described by: [ a(x - x₀) + b(y - y₀) = 0 ]

6. Point-Normal Equation of a Plane

• Plane containing point P₀(x₀, y₀, z₀) with normal vector n = (a, b, c):

  • [ a(x - x₀) + b(y - y₀) + c(z - z₀) = 0 ]

7. Forms of Line and Plane Equations

• 
  

  Theorem 3.3.1:(a) For line:

  • 
    

    If a and b are not both zero, ax + by + c = 0 defines a line with normal n = (a, b).(b) For plane:

  • If a, b, and c are not all zero, ax + by + cz + d = 0 describes a plane in R³ with n = (a, b, c).

8. Vector Orthogonality

Example #3

(a) Show vector n₁ = (a, b) in ax + by = 0 is orthogonal to the line.(b) Show vector n₂ = (a, b, c) in ax + by + cz = 0 is orthogonal to the plane.

9. Vector Decomposition

• Vectors can be decomposed into linear combinations of other vectors.

• Example: [ u = w₁ + w₂ ] where:

  • w₁ is parallel to a vector a

  • w₂ is orthogonal to a

10. Orthogonal Projection

• Definition:

  • w₁ = projection of u on a: written as: [ w₁ = ext{proj}_{a} u ]

  • w₂ = component of u orthogonal to a: [ w₂ = u - ext{proj}_{a} u ]

11. Calculation of Orthogonal Projection

• Formula:

  • [ ext{proj}_{a} u = rac{u ullet a}{ orm{a}^2} a ]

12. Examples of Projections

Example #4

• Given u = (2, −1, 3) and a = (4, −1, 2), find components along and orthogonal to a.

13. Norm of Projection

• The norm of the projection of u on a: [ || ext{proj}_{a} u|| ]

14. Pythagorean Theorem for Orthogonal Vectors

• If u and v are orthogonal vectors in Rⁿ, [ ||u + v||^2 = ||u||^2 + ||v||^2 ].

15. Distance Formulas

• Distance from a point to line in R²: [ D = rac{|ax_0 + by_0 + c|}{ orm{(a,b)}} ]

• Distance from a point to a plane in R³: [ D = rac{|ax_0 + by_0 + cz_0 + d|}{ orm{(a,b,c)}} ]

16. Example Distance Problems

Example #7

Find distance D between point (1, −4, −3) and the plane 2x − 3y + 6z = −1.

Example #8

Find distance between planes x + 2y − 2z = 3 and 2x + 4y − 4z = 7.

17. Practice Problems

• Complete exercises 1, 3, 4, 5, 7, 9, 11, 13, 15, 17, 21, 23, 25, 27, 29 of Section 3.3.