Overview
This lecture introduces vectors, their properties, notation, and operations such as component resolution, addition, subtraction, and dot product in physics.
Scalars vs. Vectors
- A scalar has only magnitude (size), while a vector has both magnitude and direction.
- Examples: Distance and speed are scalars; position, displacement, velocity, and acceleration are vectors.
- Displacement and position vectors require a reference point called the origin.
Vector Notation
- Vectors are denoted with an arrow or bold font (e.g., A or (\vec{A})).
- Unit vectors ((\hat{x}), (\hat{y}), (\hat{z})) indicate direction along coordinate axes and have magnitude 1.
- Components of a vector are written as (A_x \hat{x} + A_y \hat{y} + A_z \hat{z}).
Vector Magnitude and Direction
- The magnitude of a vector is found using the Pythagorean theorem: ( |\vec{A}| = \sqrt{A_x^2 + A_y^2 (+ A_z^2)} ).
- The angle of a vector with respect to the x-axis can be found using arctangent: (\theta = \arctan(A_y / A_x)).
- Be careful with quadrant signs, as arctangent loses some directional information.
Dot Product
- The dot product of two vectors ((\vec{A} \cdot \vec{B})) projects one vector onto another.
- Calculated as: ( A_x B_x + A_y B_y (+ A_z B_z) ) or ( |\vec{A}||\vec{B}|\cos\theta ).
- Used to find the angle between a vector and an axis with unit vectors.
Resolving Vectors into Components
- Vectors can be split into x and y components using right triangles and trigonometry.
- (A_x = A \cos\theta), (A_y = A \sin\theta), with (\theta) measured from the x-axis.
- If angle is from y-axis, sine and cosine roles are swapped.
Vector Addition and Subtraction
- To add vectors graphically, place the tail of one at the tip of another; the resultant goes from first tail to last tip.
- Algebraic addition: sum corresponding components, then find magnitude and direction.
- Subtract by adding the negative of a vector.
Key Terms & Definitions
- Scalar — a quantity with magnitude only, no direction.
- Vector — a quantity with both magnitude and direction.
- Unit vector ((\hat{x}), (\hat{y}), (\hat{z})) — a vector of length one pointing along a coordinate axis.
- Dot product — operation yielding a scalar from two vectors, equals product of magnitudes and cosine of angle between.
- Component — projection of a vector along a given axis.
Action Items / Next Steps
- Practice resolving vectors into components using sine and cosine.
- Solve problems involving vector addition and subtraction both graphically and algebraically.
- Review and memorize key definitions and vector operations.