Scalars, Vectors, Tensors
The Physics Hypertextbook™
© 1998-2008 by Glenn Elert -- A Work in Progress
All Rights Reserved -- Fair Use Encouraged
Discussion
introduction
text
| Quantities Compared | ||
| aspect | scalars | vectors |
|---|---|---|
| definition | a quantity that has magnitude only |
a quantity that has both magnitude and direction |
| up, down, left, right north, east, west, south forward, backward parallel, perpendicular positive, negative (in a coordinate system) bearing angle angle of inclination, depression angle with the vertical, horizontal right ascension, declination |
||
| examples | distance mass time (classical) surface area volume density speed energy |
displacement weight proper time (relativistic) projected area velocity acceleration force momentum electric, magnetic, gravitational fields |
| mathematics | "ordinary" arithmetic addition, subtraction sum, difference multiplication |
vector arithmetic vector addition, vector subtraction resultant (net), change (delta) dot (scalar) product, cross (vector) product |
| 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 |
special triangles
| 45°, 45°, 90° 1 : 1 : √(2) |
An isosceles right triangle. The two legs are equal in size. If we assume each leg has a length of one, according to Pythagoras' theorem, the hypotenuse has a length equal to the square root of two. | |
| 30°, 60°, 90° 1 : 2 : √(3) |
Half an equilateral triangle. Assume the sides of the full triangle have a length of two. Split it in half and the side opposite the 30° angle has a length of one. Using Pythagoras' theorem, the remaining leg has a length equal to the square root of three. | |
| 37°, 53°, 90° 3 : 4 : 5 |
The simplest Pythagorean triple. The size of the angles are best determined with a calculator. Standardized tests love this triangle. (Secret reason: It simplifies grading.) Some people memorize these angles to help them spot 3-4-5 triangles on tests. |
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Summary
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Problems
practice
- Write something.
- Answer it.
- Write something.
- Answer it.
- Write something.
- Answer it.
- Write something completely different.
- Answer it.
numerical
- problems
Resources
- general
- The Mechanical Universe and Beyond (video on demand, login required)
- Vectors, Physics must explain not only why and how much, but also where and which way.
- The Mechanical Universe and Beyond (video on demand, login required)
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