Lecture Notes: Vector

Introduction

• Vector: Any quantity with direction and magnitude.
• In mathematics, vectors must adhere to the parallelogram rule for addition.

Features of Vectors

• Ability to add with direction and magnitude.
• A rule is required for combining direction and magnitude.

Types of Vectors

• Collinear Vectors: Parallel vectors.
• Non-Collinear Vectors: No two vectors are parallel.

Vector Calculation

• If A, B, C are non-parallel: A + B + C = 0

Dot Product

• A.B = |A||B|cos(θ)
• Features:
  • A.B = 0 means A and B are perpendicular.
  • (A.B) + (A.C) = A.(B+C)

Cross Product

• A x B: Perpendicular to both A and B
• Features:
  • A x A = 0
  • A x B = - (B x A)

Scalar Triple Product

• A.(B x C) = Volume of parallelepiped
• If A, B, C are coplanar, then A.(B x C) = 0

Vector Triple Product

• A x (B x C) = B(A.C) - C(A.B)
• Features:
  • Pay attention to brackets; they affect the result.

Important Points

• Vectors are defined by magnitude and direction.
• Follow correct calculations and rules.
• Recognize and apply mathematical harmonies.

Conclusion

• This lecture focused on the basics of vectors and their applications.
• It is essential to correctly understand mathematical concepts.