Vector Addition and Subtraction
Summary
- Physical quantities come in several types.
- The type of physical quantity determines the type of mathematics used.
- Only two types are dealt with in this book: scalar quantities and vector quantities — usually shortened to scalars and vectors.
| aspect | scalars | vectors |
|---|---|---|
| definition | a quantity that has magnitude only |
a quantity that has both magnitude and direction |
| up, down, left, right forward, backward north, east, west, south parallel, perpendicular radial, tangential positive, negative bearing angle angle of inclination, depression angle with respect to… right ascension, declination |
||
| examples | distance time speed mass energy surface area volume density |
displacement velocity acceleration force weight momentum projected area |
| fields | Higgs field | gravitational field*, electric field, magnetic field, electromagnetic field, electroweak fields, gluon field |
| mathematics | ordinary arithmetic addition, subtraction sum (+), difference (−) multiplication |
geometry, trigonometry, vector arithmetic vector addition, vector subtraction resultant (∑), change (∆) dot product (·), cross product (×) |
| solutions | diagrams are often helpful | diagrams are essentially mandatory |
| answers | a number with a unit | a number with a unit and a direction angle -or- a number with a unit along each coordinate axis -or- an arrow drawn to scale in a specific direction |
- Vector addition
- Parallel vectors behave like numbers on a number line.
- Add the magnitudes of vectors in the same direction.
- Subtract the magnitudes of vectors in opposite directions.
- Perpendicular vectors behave like points on a coordinate plane.
- Use Pythagorean theorem to determine magnitude.
- Use the tangent function to determine direction.
- Vectors in standard position have a common origin and are used in the parallelogram rule of vector addition
- Construct a parallelogram using two vectors in standard position.
- The resultant is the diagonal of the parallelogram coming out of the common vertex.
- Your diagram should look like a parallelogram made of vectors. (Note: When the two original vectors are perpendicular, the parallelogram will be a rectangle.)
- Vectors arranged head to tail (with the tail of the second vector placed on the head of the first) are used in the triangle rule of vector addition.
- Given two vectors arranged head to tail.
- The resultant is the vector drawn from the tail of the first to the head of the second.
- Your diagram should look like a triangle made of arrows. (Note: When the two original vectors are perpendicular, the triangle drawn will be a right triangle and the resultant will be a hypotenuse.)
- Vector addition is similar to arithmetic addition.
- Vector addition is a binary operation. (Only two vectors can be added at a time.)
- Vector addition is commutative. (The order of addition is unimportant.)
- Parallel vectors behave like numbers on a number line.