The Physics
Hypertextbook
Opus in profectus

## 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.
Physical quantities compared * The gravitational field is a vector field in classical physics and a tensor field in general relativity. A tensor is a mathematical object that is more complex than a vector, kind of like the way a vector is more complex than a scalar.
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
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
sum (+), difference (−)
multiplication
geometry, trigonometry, vector arithmetic
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
• 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
1. Construct a parallelogram using two vectors in standard position.
2. The resultant is the diagonal of the parallelogram coming out of the common vertex.
3. 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.
1. Given two vectors arranged head to tail.
2. The resultant is the vector drawn from the tail of the first to the head of the second.
3. 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.)