Overview
This lecture explains unit vectors in Cartesian coordinates, their properties, and how to express vectors using unit vectors i-hat and j-hat.
Unit Vectors in Cartesian Space
- Unit vectors have magnitude 1 and specify direction only; they are dimensionless.
- In Cartesian coordinates, the standard unit vectors are i-hat (x-axis), j-hat (y-axis), and k-hat (z-axis).
- The "hat" notation (e.g., î) indicates a unit vector.
Writing Vectors with Unit Vectors
- Any vector can be written as a combination of unit vectors along the axes.
- For example, a vector from the origin to point (5, 4) is written as 5 î + 4 ĵ.
- The coefficients in front of î and ĵ represent the components (aₓ and aᵧ) along the x and y axes, respectively.
- To find the components, count the distance along each axis from the origin to the point.
Key Terms & Definitions
- Unit vector — a vector with magnitude 1 and no units, used to specify direction.
- i-hat (î) — unit vector in the x-direction.
- j-hat (ĵ) — unit vector in the y-direction.
- k-hat (k̂) — unit vector in the z-direction.
- Component — the part of a vector along a particular axis (e.g., aₓ along x).
Action Items / Next Steps
- Practice writing vector components using unit vectors for given points in Cartesian coordinates.