Overview
This lecture introduces vectors and their operations, covering coordinate systems, vector notation, scalar and vector operations, and both dot and cross products.
Coordinate Systems in Two Dimensions
- The x-axis is horizontal and the y-axis is vertical, forming the 2D coordinate system.
- Points are located by tracing lines parallel to axes and recording x and y coordinates, always written (x, y).
- The origin, labeled O, is where the axes cross and has coordinates (0, 0).
Vectors: Definition and Notation
- Vectors are arrows defined by length (magnitude) and direction, not by position.
- Components of a vector indicate steps in x (horizontal) and y (vertical) directions.
- Components are written in square brackets: (\begin{bmatrix}x \ y\end{bmatrix}).
- Vectors can be generalized to n dimensions, with components listed from 1 to n.
Scalar Operations with Vectors
- Scalar operations involve multiplying or dividing a vector by a number (scalar).
- Each component is multiplied or divided by the scalar.
- Multiplying by a negative scalar reverses the vector’s direction.
Vector Operations: Addition and Subtraction
- Vector addition: add corresponding x and y components.
- Graphically, place tails together; the result is an arrow from the origin to the new tip.
- Subtraction gives a vector from the tip of one to the tip of the other and can represent the difference between two points.
Magnitude (Length) and Unit Vectors
- The magnitude (norm) of a vector in 2D: ( \sqrt{x^2 + y^2} ).
- A unit vector has a length of 1 and is found by dividing the vector by its own length.
- Standard unit vectors: (\hat{x}), (\hat{y}), and (\hat{z}) for their respective axes.
Dot Product (Scalar Product)
- The dot product of vectors A and B: (A \cdot B = |A||B|\cos(\theta)), where θ is the angle between them.
- If vectors are parallel: dot product is max; perpendicular: dot product is 0; opposite: dot product is negative max.
- Measures similarity of direction between vectors.
- The length of a vector can be found as the square root of the dot product with itself.
Cross Product (Vector Product)
- The cross product, (A \times B), is a vector perpendicular to both A and B, only defined in 3D.
- Result depends on the order (not commutative): (A \times B = - (B \times A)).
- Magnitude: (|A||B|\sin(\theta)), where θ is the angle between the vectors.
- Zero if vectors are parallel, maximum if perpendicular.
- Direction follows the right-hand rule.
- Length of the cross product gives the area of the parallelogram formed by the vectors.
Key Terms & Definitions
- Coordinate System — A grid defined by perpendicular axes (x, y) for specifying points.
- Vector — An arrow with magnitude and direction, represented by components.
- Scalar — A real number used to scale a vector.
- Magnitude/Norm — The length of a vector.
- Unit Vector — A vector with magnitude 1, indicating direction only.
- Dot Product — Scalar result indicating how parallel two vectors are.
- Cross Product — Vector result perpendicular to two 3D vectors, indicating how perpendicular they are.
Action Items / Next Steps
- Practice calculating vector components and operations.
- Study the properties and formulas for dot and cross products.
- Review matrix determinants for understanding cross product computation in detail.