Overview
This lecture introduces scalars and vectors, essential concepts for understanding quantities and operations in physics, and explains basic vector operations and applications.
Scalars and Vectors
- Scalars are quantities described by magnitude only, such as mass, time, or temperature.
- Vectors have both magnitude and direction, answering "how much" and "which way."
- Vectors are represented by bold letters with arrows on top in notation.
Vector Representation and Operations
- The direction of a vector is as important as its magnitude.
- Vector addition: place vectors head to tail and draw the resultant from the start of the first to the end of the second.
- If vectors are aligned, the resultant's magnitude is their sum.
- If vectors are at an angle, the resultant forms the hypotenuse of a right triangle.
- The Pythagorean theorem is used to calculate the magnitude of the resultant: ( C = \sqrt{A^2 + B^2} ).
Trigonometric Functions with Vectors
- Right triangles are labeled with opposite, adjacent, and hypotenuse sides relative to an angle theta (( \theta )).
- SOHCAHTOA helps remember: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent.
- To find the angle, use the inverse trigonometric functions (e.g., ( \theta = \sin^{-1}(3/5) )).
- Calculators can use degrees or radians for angle measurement.
Other Vector Operations
- Vector subtraction: invert the direction of the second vector and add as usual.
- A vector can be split into X and Y components using trigonometry, useful for projectile motion analysis.
- Multiply a vector by a scalar by changing the magnitude only, keeping direction constant.
Example: Boat in a River
- For a boat moving across a river, draw vectors for boat speed (horizontal) and current (vertical).
- The resultant velocity is found using the Pythagorean theorem: ( \sqrt{4^2 + 1^2} = \sqrt{17} ).
- To find direction: ( \theta = \tan^{-1}(1/4) = 14^\circ ) off the horizontal.
Key Terms & Definitions
- Scalar — a quantity with magnitude only.
- Vector — a quantity with both magnitude and direction.
- Resultant Vector — the sum of two or more vectors.
- Component — a part of a vector along an axis (e.g., X or Y).
- SOHCAHTOA — mnemonic for trigonometric ratios in right triangles.
Action Items / Next Steps
- Review scientific notation, units, and dimensional analysis if unfamiliar.
- Practice basic vector addition, subtraction, and splitting into components.
- Ensure calculator is set to the correct angle mode (degrees for now).
- Prepare for upcoming lessons on trigonometric functions and more advanced vector operations.