Vector and Scalar Quantities
Scalars
Every measurable quantity has a numerical value associated with it, which
is called the "magnitude" of the quantity. There will also be a unit of
measurement. If this is all that is needed to completely describe the
measurement, then the quantity is known as a scalar quantity. Some
examples are energy, time, price, temperature, pressure and area.
Other quantities have an associated direction that must also be given in
order to completely define the quantity. These are called vector
quantities. Examples are force, momentum and velocity. Another type of
quantity is the phasor.
Representing vectors and scalars
In type, as well as handwritten text, scalar quantities are represented
by letters in normal format. They may be upper or lower case, Roman or
Greek. P, G, x, d, h, µ, 0.
In type, vectors are usually represented by bold, italic symbols. The ~
symbol is used under a handwritten symbol for a vector. F, v, a, p.
~
Similar vector and scalar quantities
Sometimes, whether or not a direction has to be considered depends on the
situation, or may be a matter of opinion. This is particularly the case when
considering movement. There are two versions of some quantities.
"Distance" is a scalar, measuring how far something has travelled.
"Displacement" is a vector, measuring how far an end point is from the
starting point, and in what direction ("as the crow flies.")
"Speed" is a measure of how much ground is being covered in a set time.
"Velocity" measured how far you travel in a particular direction per unit of
time.
"Acceleration" may be a scalar or a vector, but we do not have separate names
for the two.
25th St
24th St
23rd St
H
22nd St
Example: You are staying at a hotel
in Manhattan (H on the "map". You
walk 3 blocks north, then 2 blocks
west, then 3 blocks south. The
"distance" you have walked is 8
blocks. Your "displacement" from
H the hotel is 2 blocks west.
Vector Diagrams
It is often necessary to draw diagrams to represent vector quantities.
Each vector is represented by a "directed line segment" - an arrow.
The length of the arrow represents the magnitude of the vector (so there
must be a scale for the diagram.)
The direction of the arrow represents the direction of the vector (so there
must be a reference direction or directions on the diagram.)
The universe, as we perceive it, is three-dimensional. However, physics
problems are often restricted to one or two dimensions. In a 1-D problem, the
only directions will be forwards and backwards, or up and down. Frequently +
and - will be enough to indicate which is which.
In a 2-D problem, the directions will usually be vertical and horizontal or the
compass directions.
Compass Directions
Use these names for
exact directions only.
Measure bearings
clockwise from north.
Give the values in 3 figures
e.g. West is 270°, NE is 045°
Vector Arithmetic
Calculations with scalar quantities can be carried out using normal
arithmetic. However, a geometric method is needed with vectors, in order
to take account of the directions.
Multiplying a vector by a scalar
No change of direction is involved. The magnitude of the vector is
multiplied by the scalar. On a vector diagram, the arrow changes length,
but not direction.
Adding vectors
Place the vectors head-to-tail on the diagram. The sum (resultant vector)
is found by drawing an arrow from the beginning of the first vector to
the end of the last. Note that vector diagrams do not take account of the
location of the vectors. (On hand drawings, the resultant is conventionally
indicated with a double arrowhead.)
a
b+a
b
3b
a+b
Subtracting vectors
The simple rule for subtracting vectors is "add the opposite",
so a - b becomes a + -b
so we have
a
b
-b
Note that a - b = -(b - a)
-a
Resolving vectors into components
Sometimes 2-D vector problems can be simplified by splitting them into
1-D problems. Some vectors must then be split into two separate parts
of which the original vector is the resultant. The components will
normally be in two perpendicular directions. This is often vertical and
horizontal, or sometimes "parallel to the surface" and "perpendicular to
the surface."
e.g. a ball is thrown with a velocity
of 20 ms-1 at an angle of 30° above
the horizontal.
The horizontal component of this
velocity is 20 cos 30° or 17 ms-1 .
The vertical component if this
velocity is 20 sin 30° or 10 ms-1.
These problems are much easier if you are familiar with the ratios
of the most common triangles, 3-4-5, 30°- 60°- 90° and 45°- 45°- 90°
Triangles
45 degree triangle
sin 45° = 1/√2
cos 45° = 1/√2
tan 45° = 1
30-60-90° triangle
sin 30° = 1/2
cos 30° = √3/2
tan 30° = 1/√3
sin 60° = √3/2
cos 60° = 1/2
tan 60°= √3
3-4-5 triangle
sin 37° = 3/5 = 0.6
cos 37° = 4/5 = 0.8
tan 37° = 3/4 = 0.75
sin 53° = 4/5 = 0.8
cos 53° = 3/5 = 0.6
tan 53° = 4/3 = 1.33