Overview
This lecture covers representing vectors using unit vectors and components, finding vector components with trigonometry, and performing vector arithmetic using components.
Unit Vectors
- Unit vectors have a length of one and no physical units, indicating only direction.
- Standard unit vectors are î (x-direction), ĵ (y-direction), and k̂ (z-direction).
- In most physics problems, focus is on î and ĵ for motion in a plane.
- Any vector can be constructed by multiplying unit vectors by scalar quantities and adding them.
Constructing Vectors with Components
- Any vector (e.g., velocity) can be written as a sum of its x and y components:
( \vec{v} = v_x \hat{i} + v_y \hat{j} )
- The numbers (like ( v_x ), ( v_y )) are vector components, giving how much to multiply each unit vector.
Finding Components Using Trigonometry
- Given a vector’s magnitude and angle, use:
- ( v_x = v \cos \theta )
- ( v_y = v \sin \theta ) (adjust the sign as needed for direction)
- Angles are often kept between 0° and 90° for calculator consistency; assign signs manually.
Magnitude and Direction from Components
- Magnitude: ( v = \sqrt{v_x^2 + v_y^2} )
- Direction: ( \theta = \arctan\left(\frac{|v_y|}{v_x}\right) ), specify quadrant and signs for direction (e.g., 'south of east').
Vector Arithmetic with Components
- Vectors can be written as ordered pairs: (x, y), omitting unit vectors.
- To add or subtract vectors, add or subtract corresponding components.
- Multiplying a vector by a scalar multiplies each component.
Coordinate System Choices
- Coordinates (x, y) can be aligned for convenience, e.g., x-axis parallel to an incline.
- For inclined axes, components may be labeled parallel (( a_{\parallel} )) and perpendicular (( a_{\perp} )).
Key Terms & Definitions
- Unit vector — vector of length one with no units, indicating direction.
- î, ĵ, k̂ — unit vectors in x, y, z directions, respectively.
- Component — the scalar amount a vector has in a given direction.
- Magnitude — the length of a vector.
- Ordered pair — a vector represented as (x, y) components.
Action Items / Next Steps
- Practice breaking vectors into components using trigonometry.
- Prepare to use vector components in multi-dimensional kinematics problems.