While studying physics, there are many different aspects to measure and many types of measurement tools, where scalar and vector quantities are two the these types of measurement tools (Gunner, 2021). Or it can be said that vector and scalar quantities are the types of physical quantities that are used in physics (Tutorialspoint, 2022). Before jump to the definitions, let discuss in brief about what the difference between displacement and distance. The fact that displacement is defined by both direction and magnitude, while distance is defined only by magnitude, explains why displacement is an example of a vector quantity, while distance is an example of scalar quantity (OpenStax, 2022). A short introduction of scalar and vector quantities with example s are given here.
definition Link to heading
Scalar quantities are quantities that only have a magnitude or size associated with them, can be expressed completely with single number, and are operated with usual rules of arithmetic, while on the other hand, vector quantities are quantities that have also a direction associated with them beside the magnitude, have components, and requires vector operations as part of algebra (anjalishukla1859, 2022).
examples of scalar quantities Link to heading
Mass density, mass, and volume are all scalar quantities and are related to each other through
$$ \rho = \frac{m}{V}. $$
Charge density, charge, and volume are also scalar quantities and their relation is as follow
$$ \rho = \frac{q}{V}. $$
Charge, capacitance, and voltage, which all are scalar quantities have following relation
$$ q = CV. $$
And for rigid body that performs rotation it has
$$ I = cML^2, $$
which are moment of inertia, a coefficient, total mass, and a particular length of the object. Remember also ideal gas law
$$ p = \frac{nRT}{V} $$
There might also a scalar quantity that is result of product of scalar quantity and dot product of vector quantity
$$ K = \tfrac12 m v^2 = \tfrac12 m (\vec{v} \cdot \vec{v}) $$
as for relation between kinetic energy, mass, and velocity. And also
$$ p = \vec{F} \cdot \vec{A}, $$
that relates pressure, force, and area. Notice that area is a pseudovector since it can be defined by two non-parallel vectors (Wikipedia, 2023).
examples of vector quantities Link to heading
Relative position is example of a vector quantity
$$ \vec{r}_{ab} = \vec{r}_a - \vec{r}_b, $$
which is the difference between object position and a reference position. It can also result of product a scalar quantity and a vector quantity
$$ \vec{F} = m \vec{a}. $$
that relates force, mass, and acceleration. Electric field is also similar
$$ \vec{F}_E = q \vec{E}. $$
and magnetic field is product of scalar and cross product
$$ \vec{F} = q \vec{v} \times \vec{B}. $$
As a pseudovector area can be obtained using
$$ \vec{A} = \vec{l} \times \vec{w}, $$
which is a cross product of length and width.
presentation Link to heading
A physical quantity usually has symbol, numerical value, and unit, where the numerical value is often displayed in scientific notation. For a scalar quantity it will be displayed as
$$ q = 1.602 \times 10^{-9} \ {\rm C} = 1.602 \ {\rm nC}, $$
while for a vector quantity it migh be displayed as
$$ \begin{array}{rcl} \vec{N} & = & (3 \ \hat{i} + 4 \ \hat{j} + 12 \ \hat{k}) \times 10^{-6} \ {\rm N} \newline & = & 13 \ (\tfrac{3}{13} \ \hat{i} + \tfrac{4}{13} \ \hat{j} + \tfrac{12}{13} \ \hat{k}) \ {\rm \mu N} \end{array} $$
since it has vector components in x, y, z directions. In the first line vector is displayed as components and in second line it is displayed as magnitude and direction.
Notice also that in two previous equations there are n and μ metric or SI prefixes used that stand for 10⁻⁹ and 10⁻⁶ (NIST, 2010).
challenges Link to heading
- Find the 24 metric prefixes and write the order of 10 they represent.
- Represent a vector (40 $\hat{i}$, 60 $\hat{j}$) in magnitude and direction.
- Represent the value 1.602 nC in μC and mC.
- What would be the result of force magnitude of charge 1 mC in an electric field with magnitude of 0.2 μN/C? Represent the result in scientific notation first and then use a metric prefix that give the most compact form.
- What is the difference between speed and velocity? Which one is vector quantity? How the two quantities are related?
- What is the difference between relative position, displacement, and distance?
- Relate also the explanation with scalar and vector quantities. Volume can be obtained from area and height. What would it be? A scalar or vector quantity?
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