Lecture on Linear Systems and Orthogonal Vectors

Introduction

• Presenter: Nathan Johnson
• Series: Using linear systems to solve problems
• Focus of this video: Finding vectors orthogonal to a given set of vectors

Finding Orthogonal Vectors in 2D

• Given vector: ( \mathbf{v} = (2, 1) )
• Objective: Find a vector orthogonal to ( \mathbf{v} )
  • Original vector slope: ( \frac{1}{2} )
  • Orthogonal vector slope: Negative reciprocal -> (-2)
  • Example orthogonal vector: ( (-1, 2) )

Orthogonal Vectors in 3D

• Given vectors: ( \mathbf{a} = (1, 2, 3) ), ( \mathbf{b} = (-1, -3, 3) )
• Objective: Find a vector orthogonal to both
  • Common method: Cross product (not preferred by lecturer)
  • Alternative: Linear systems

Solving via Linear Systems

• Define vector ( \mathbf{v} = (x, y, z) )
• Equations from orthogonality:
  • ( x + 2y + 3z = 0 )
  • (-x - 3y + 3z = 0 )

• Matrix form: [ \begin{bmatrix} 1 & 2 & 3 \ -1 & -3 & 3 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix}

  \begin{bmatrix} 0 \ 0 \end{bmatrix} ]
• Perform row reduction to solve
• Find free variable (e.g., (z = 1))
• Solve for leading variables: ( x = -15, y = 6 )
• Resulting orthogonal vector: ( (-15, 6, 1) )

Orthogonal Vectors in 4D

• Given vectors: ( (1, 2, 2, 2), (2, 1, -2, 1), (1, 0, 2, 0) )
• Objective: Find a vector orthogonal to all
• Define ( \mathbf{v} = (v_1, v_2, v_3, v_4) )
• Equations from orthogonality
  • Solve via Gaussian elimination to reduced row echelon form
  • Identify free variable (e.g., (v_4))
• Solve for leading variables
• Example solution: ( (0, -1, 0, 1) )

Higher Dimensional Spaces and Multiple Orthogonal Vectors

• Given vectors in 4D
  • Objective: Describe all vectors orthogonal to ((1, 2, 3, 4), (2, 1, 3, 2), (-2, 2, 0, 4))
• Observe two free variables leading to a plane of solutions
• General solution:
  • (v_1 = -v_3)
  • (v_2 = -v_3 - 2v_4)
• Solution space description is a linear combination of two vectors:
  • ((-1, -1, 1, 0)) and ((0, -2, 0, 1))

Conclusion

• Infinite solutions possible in higher dimensions
• Future videos will cover more problems solvable via linear systems

Key Takeaway: Linear systems provide a dimension-independent method of finding orthogonal vectors across different spaces, offering an alternative to operations like the cross product.