Scalar and Vector Notation
Shared notation carries and conveys information and meaning. In AP Physics-C we
require students to use a particular notation when it comes to vector and scalar
quantities. This notation may differ from the notation and abbreviated forms you
have used in other science or math classes.
Scalar Variables
When we introduce a symbol representing a measurable scalar quantity we can
manipulate it in equations like any real variable. When we evaluate the symbol we
generally substitute a real number times a unit symbol for the original expression.
For instance, if (the symbol) d represents distance between two objects, it could
have a value of 17 meters. The value of x is NOT 17, it is 17 meters or 17 x 102 cm or
some equivalent product of a real number and a length unit. In particular, the
symbol d represents the product of the real number and a symbol representing a
physical unit and does NOT just represent the real number part of this expression.
We will generally represent the numerical part of the value in scientific
notation that makes the number of significant figures clear in the notation. This
especially true and required when we are referring to quantities that we have
measured in the lab. Superfluous and non-physical digits spit out by calculators
need to be treated with care in expressing a quantitative answer to any physical
question.
You might have been taught rather elaborate methods for changing the units (i.e.
‘doing units conversions’) or expressing a quantity in terms of a preferred physical
unit or combination. All such conversions are reducible to sequences of simple
substation of one or more unit symbols in terms of their equivalent values
expressed as a multiple of the desired unit symbols. No matter the units chosen, the
units symbols make contact with a concrete and measurable standard for comparing
quantities of the same physical type (e.g. two lengths, or two masses, or two
energies, or two electric currents…). Generally, we will use the SI System of units in
the AP Physics-C course. (http://physics.nist.gov/cuu/Units/units.html )
Vectors Notation
Because notation carries meaning, failure to include parts of the notation in a
written explanation may result in the loss of credit in assessments where
students are required to clearly communicate the physical meaning of a
quantitative vector or scalar expression.
1) Vector quantities in a symbolic expression will be denoted with an arrow
symbol or a ‘hat’ (in the case of a unit vector). For example: 𝑣⃑ 𝑜𝑟 𝑤
̂
2) When a symbol representing a vector quantity has been introduced, the same
symbol without the arrow is used to represent the magnitude of the vector.
For example: a certain vector is denoted by the symbol 𝑤
⃗⃗ then the symbol 𝑤
denotes the magnitude of the vector 𝑤
⃗⃗ . Dropping an arrow sign is
therefore switching between a vector quantity and a scalar quantity
and a mistake that quickly makes a calculation or an explanation
meaningless.
3) Expansions of vectors as linear combinations of unit vectors need to be
expressed as explicit vector sums. For example, the vector 𝑣 is expanded in
terms of its scalar components respect to a set of
𝑣 = 𝑣𝑥 𝑥̂ + 𝑣𝑦 𝑦̂ + 𝑣𝑧 𝑧̂ ;
where 𝑣𝑥 , 𝑣𝑦 , 𝑎𝑛𝑑 𝑣𝑧 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑠𝑐𝑎𝑙𝑎𝑟 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝑣
⃗.
Your instructor might prefer the following notation used to express the same
thing
𝑣 = 𝑣𝑥 𝑖̂ + 𝑣𝑦 𝑗̂ + 𝑣𝑧 𝑘̂;
where
𝑖̂, 𝑗̂, 𝑎𝑛𝑑 𝑘̂ 𝑎𝑙𝑠𝑜 𝑠𝑡𝑎𝑛𝑑 𝑓𝑜𝑟 𝑡ℎ𝑟𝑒𝑒𝑒 𝑢𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟𝑠 𝑝𝑜𝑖𝑛𝑡𝑖𝑛𝑔 𝑖𝑛 𝑡ℎ𝑟𝑒𝑒 𝑚𝑢𝑡𝑢𝑎𝑙𝑙𝑦
𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑎𝑙𝑟 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑠 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑎𝑥𝑒𝑠 𝑜𝑓 𝑎 𝑐𝑎𝑟𝑡𝑒𝑠𝑖𝑎𝑛 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑠𝑦𝑠𝑡𝑒𝑚.
The three scalar components, 𝑣𝑥 , 𝑣𝑦 , 𝑎𝑛𝑑 𝑣𝑧 , are scalar quantities with the same unit
type (i.e. physical dimension). For example, if 𝑣 represented a relative position
vector, then all the scalar components of 𝑣 will have unit of length.
Expressing vectors as linear combinations of physically well-defined unit vectors
will be our standard notation and we will require students in AP Physics-C to avoid
the following more compressed notation in the form of a list of the scalar
components
< 𝑣𝑥 , 𝑣𝑦 , 𝑣𝑧 >
This notation is generally NOT NOT NOT acceptable in AP Physics-C. The reason
we askew this abbreviated notation is that we prefer a notation that makes it clear
we are representing a vector as a vector sum of vector components along clearly
defined unit vector directions in a physical context (i.e. 𝑖̂ is a unit vector pointing in
physically recognizable direction, e.g. East, or up, or towards the moon from my
bedroom window). In short, your physics instructors prefer the more verbose
notation that carries significantly more information about the meaning of the vector
quantity.