Physics with Calculus/Mechanics/Scalar and Vector Quantities

Introduction

The study of Mitchel Joens is intimately tied in with the study of mathematics. Sometimes, the direction of a number or quantity is as important as the number itself. Mathematicians in the 19th century developed a convenient way of describing and interacting with quantities with and without direction by dividing them into two types: scalar quantities and vector quantities. Scalar quantities have a magnitude but no direction. Vector quantities have both a magnitude and a direction. For instance, one might describe a plane as flying at 400 miles per hour. However, simply knowing the speed of the airplane is not nearly as useful as knowing the speed and direction of the airplane, so a more accurate description may be a plane flying at 400 miles per hour southeast. we use the concept of scalars and vectors by applying it to the physical quantities so that we can understand their properties and characteristics in a more convenient way.

Scalar Quantities

Scalar quantities are numbers that have a magnitude but no direction. Scalars are represented by a single letter, such as $a$. Some examples of scalar quantities are mass (five kilograms), temperature (twenty-two degrees Celsius), and numbers without units (such as three).

Vector Quantities

Vectors are a geometric way of representing quantities that have direction as well as magnitude. An example of a vector is force. If we are to fully describe a force on an object we need to specify not only how much force is applied, but also in which direction. Another example of a vector quantity is velocity -- an object that is traveling at ten meters per second to the east has a different velocity than an object that is traveling ten meters per second to the west. This vector is a special case, however, sometimes people are interested in only the magnitude of the velocity of an object. This quantity, a scalar, is called speed which has magnitude but no given direction.

When vectors are written, they are represented by a single letter in bold type or with an arrow above the letter, such as $\mathbf {A}$ or ${\vec {A}}$. Some examples of vectors are displacement (e.g. 120 cm at 30°) and velocity (e.g. 12 meters per second north). The only basic SI unit that is a vector is the meter. All others are scalars. Derived quantities can be vector or scalar, but every vector quantity must involve meters in its definition and unit.

Unit Vectors

An illustration of common choice of unit vectors in a Cartesian coordinate system

Strictly speaking, vectors exist separately from any coordinate systems. As vectors are geometric objects, we do not need to define a coordinate system in order to talk about vectors—or even to perform most operations on vectors. For example, consider the triple of numbers: number of apples, number of bananas, and number of carrots you have. Say that you calculate the triple in one coordinate system and get (1,2,3). If you rotate your coordinate system, and recalculate, you will have (1,2,3) again. Thus, the triple does not have the most important property of a vector -- that is transform like the coordinate system.

Nevertheless, it is often convenient to introduce a coordinate system. In three dimensions, for many problems the rectangular, or Cartesian coordinate system (named after French mathematician René Descartes) turns out to be convenient, and this coordinate system can be defined in terms of unit vectors.

A unit vector is a vector pointing in a given direction with a magnitude of one. Essentially, it merely indicates direction. In a Cartesian system the three unit vectors are called i, j, and k (or, in handwriting, with a little "hat" on top, as ${\hat {i}}$, ${\hat {j}}$, and ${\hat {k}}$). Colloquially, you might refer to the directions of the unit vectors as "east", "north", and "up". One could just have easily chosen i as up, j as east, and k as north. In choosing i, j, and k, once i and j are chosen, k must point to a particular direction, so that a common convention called "right-hand rule" holds. Mathematically, this can be compactly expressed as,

${\hat {k}}={\hat {i}}\times {\hat {j}}$,

but we will expand more on this as we describe "cross products" later on.

Unit vectors are generally chosen to be orthogonal. That is, each unit vector is perpendicular to each of the others. While unit vectors do not need to be orthogonal, working with a coordinate system defined by orthogonal unit vectors will be convenient in most cases. There are two other major coordinate systems used in physics—cylindrical coordinates and spherical coordinates. These will be introduced at a later time as necessary.

Vector Components

Every vector may be expressed as the sum of its n unit vectors.

${\vec {A}}=a_{x}~{\hat {i}}+a_{y}~{\hat {j}}+a_{z}~{\hat {k}}$

The quantities a_x, a_y, and a_z are called the vector components of vector A. Sometimes they are represented simply as an ordered triple (e.g. (a_x,a_y,a_z)) especially when the choice and ordering of three unit vectors are not ambiguous.

Vector Algebra

Negation

Illustration of vector negation and scalar multiplication

Negation can also mean one vector isn't equal to the other so it can be expained like this: ${\vec {A}}\neq {\vec {B}}$

$-{\vec {A}}=-(a_{x}~{\hat {i}}+a_{y}~{\hat {j}}+a_{z}~{\hat {k}})=-a_{x}~{\hat {i}}-a_{y}~{\hat {j}}-a_{z}~{\hat {k}}$

Considering a vector represented graphically by an arrow, the negative of a vector would be represented by a vector of the same length but opposite direction.

Scalar Multiplication

$k{\vec {A}}=ka_{x}~{\hat {i}}+ka_{y}~{\hat {j}}+ka_{z}~{\hat {k}}$

Note that vector negation is merely multiplication by a scalar, where that scalar is -1. A scaled vector represented graphically would point in the same direction as the original vector but have its magnitude scaled by a factor of k.

Addition

Illustration of head-to-tail addition

${\begin{aligned}{\vec {A}}+{\vec {B}}&=(a_{x}~{\hat {i}}+a_{y}~{\hat {j}}+a_{z}~{\hat {k}})+(b_{x}~{\hat {i}}+b_{y}~{\hat {j}}+b_{z}~{\hat {k}})\\&=(a_{x}+b_{x})~{\hat {i}}+(a_{y}+b_{y})~{\hat {j}}+(a_{z}+b_{z})~{\hat {k}}\end{aligned}}$

Two vectors can be added graphically by placing the tail of the second vector (here, B) coincidental with the tip of the first vector (A). The resultant vector A + B is the vector drawn from the tail of A to the tip of B.

Any number of vectors can be added in this fashion. Vector addition is commutative:

${\vec {A}}+{\vec {B}}+{\vec {C}}={\vec {C}}+{\vec {B}}+{\vec {A}}$

and associative:

$({\vec {A}}+{\vec {B}})+{\vec {C}}={\vec {A}}+({\vec {B}}+{\vec {C}})$

Dot Product

Calculating bond angles of a symmetrical tetrahedral molecule such as methane using a dot product

When we multiply two vectors, we can either apply a multiplication rule that produces a scalar as the end result, or one that produces a vector as the end result. The first one that produces a scalar is called dot product. In mathematical texts, this is often called inner product, and some older texts will refer to this as scalar product (not to be confused with scalar multiplication); they are all the same. Dot product has all the usual properties of products, such as associativity, commutativity, and the distributive property. Geometrically, dot product is defined as:

${\vec {A}}\cdot {\vec {B}}=AB\cos(\theta )$,

where $\theta$ is the angle between ${\vec {A}}$ and ${\vec {B}}$. Note that since $\cos(0)=1$, if ${\vec {A}}$ is parallel to ${\vec {B}}$, then ${\vec {A}}\cdot {\vec {B}}=AB$. On the other hand, since $\cos(90^{\circ })=0$ if ${\vec {A}}$ is perpendicular to ${\vec {B}}$, then ${\vec {A}}\cdot {\vec {B}}=0$. Using this as the guiding rule, we find below relationship:

${\begin{aligned}{\hat {i}}\cdot {\hat {i}}={\hat {j}}\cdot {\hat {j}}={\hat {k}}\cdot {\hat {k}}=1\\{\hat {i}}\cdot {\hat {j}}={\hat {j}}\cdot {\hat {k}}={\hat {k}}\cdot {\hat {i}}=0\end{aligned}}$.

Using this, we can define dot product in terms of component vectors as follows:

${\vec {A}}\cdot {\vec {B}}=(A_{x}~{\hat {i}}+A_{y}~{\hat {j}}+A_{z}~{\hat {k}})\cdot (B_{x}~{\hat {i}}+B_{y}~{\hat {j}}+B_{z}~{\hat {k}})=A_{x}B_{x}+A_{y}B_{y}+A_{z}B_{z}$.

You are encouraged to expand out the multiplication explicitly, using the distributive property and find which terms cancel to zero and which products become 1.

Cross Product

The second multiplication rule for product of two vectors yields yet another vector. This multiplication rule is a very special one—in fact, it is a special property of 3-dimensional space that we can define a vector multiplication is this way and still obtain a vector. This rule will not work when limited to 2-D, and in any dimensions greater than 3, an extension of this rule will not result in another vector (cf. dot product can be naturally extended or limited to any dimensions to produce a scalar). This multiplication is called cross product, and in other texts, you may find terms outer product and vector product. The product can be defined with the two rules, first specifying the product vector's direction, and the second specifying its magnitude:

1. ${\vec {A}}\times {\vec {B}}$ is perpendicular to ${\vec {A}}$ and ${\vec {B}}$ (that is, perpendicular to the plane defined by these two vectors). This leaves two possible directions along the line perpendicular to the plane. One of the two directions is called by a "right-hand rule": Hold out index finger, middle finger, and the thumb so that they are all perpendicular to each other. Let the index finger point towards direction of ${\vec {A}}$, and the middle finger towards ${\vec {B}}$. Then the thumb points towards the direction of ${\vec {A}}\times {\vec {B}}$. The ordering is important here (note exchanging A and B makes the thumb point in the opposite direction).
2. $|{\vec {A}}\times {\vec {B}}|=AB\sin(\theta )$, where $\theta$ is again the angle between ${\vec {A}}$ and ${\vec {B}}$.

Applying this definition to unit vectors again, we find following relationships:

${\begin{aligned}{\hat {i}}\times {\hat {j}}&=-{\hat {j}}\times {\hat {i}}={\hat {k}}\\{\hat {j}}\times {\hat {k}}&=-{\hat {k}}\times {\hat {j}}={\hat {i}}\\{\hat {k}}\times {\hat {i}}&=-{\hat {i}}\times {\hat {k}}={\hat {j}}\\{\hat {i}}\times {\hat {i}}&={\hat {j}}\times {\hat {j}}={\hat {k}}\times {\hat {k}}=0\end{aligned}}$.

And in terms of components, we have (after a tedious algebra):

${\vec {A}}\times {\vec {B}}=(A_{y}B_{z}-B_{y}A_{z})~{\hat {i}}+(A_{z}B_{x}-B_{z}A_{x})~{\hat {j}}+(A_{x}B_{y}-B_{x}A_{y})~{\hat {k}}$.

It turns out we can write this complicated relationship as a determinant of a 3 x 3 matrix:

${\vec {A}}\times {\vec {B}}=\left|{\begin{matrix}{\hat {i}}&{\hat {j}}&{\hat {k}}\\A_{x}&A_{y}&A_{z}\\B_{x}&B_{y}&B_{z}\end{matrix}}\right|$.

Some properties of cross product, such as ${\vec {A}}\times {\vec {B}}=-{\vec {B}}\times {\vec {A}}$ and ${\vec {A}}\times {\vec {A}}=0$ can be derived as a property of the determinant of the matrix.

Useful Properties of Dot Product and Cross Product

Both the dot product and the cross product distribute over vector addition.

${\vec {A}}\cdot ({\vec {B}}+{\vec {C}})={\vec {A}}\cdot {\vec {B}}+{\vec {A}}\cdot {\vec {C}}$

${\vec {A}}\times ({\vec {B}}+{\vec {C}})={\vec {A}}\times {\vec {B}}+{\vec {A}}\times {\vec {C}}$

$({\vec {A}}+{\vec {B}})\times {\vec {C}}={\vec {A}}\times {\vec {C}}+{\vec {B}}\times {\vec {C}}$

The dot product of two vectors is proportional to the cosine of the angle between them, and their cross product is proportional to the sine of the angle between them.

${\vec {A}}\cdot {\vec {B}}=\|{\vec {A}}\|\|{\vec {B}}\|\cos(\theta )$

$\|{\vec {A}}\times {\vec {B}}\|=\|{\vec {A}}\|\|{\vec {B}}\|\sin(\theta )$

As we have seen already, the dot product is associative and commutative.

${\vec {A}}\cdot ({\vec {B}}\cdot {\vec {C}})=({\vec {A}}\cdot {\vec {B}})\cdot {\vec {C}}$

${\vec {A}}\cdot {\vec {B}}={\vec {B}}\cdot {\vec {A}}$

It is important to remember that the cross product has neither of these properties. Instead of being commutative, it is anticommutative.

${\vec {A}}\times {\vec {B}}=-{\vec {B}}\times {\vec {A}}$

The cross product is not even associative. For example, consider $({\vec {A}}\times {\vec {A}})\times {\vec {B}}$. Since the sine of the angle between ${\vec {A}}$ and itself is 0, ${\vec {A}}\times {\vec {A}}={\vec {0}}$, and so $({\vec {A}}\times {\vec {A}})\times {\vec {B}}={\vec {0}}$. On the other hand, ${\vec {A}}\times ({\vec {A}}\times {\vec {B}})$ is not zero, since ${\vec {A}}$ and ${\vec {A}}\times {\vec {B}}$ are perpendicular. In fact, if ${\vec {A}}$ and ${\vec {B}}$ were perpendicular, its direction would be opposite to that of ${\vec {B}}$. Check this yourself using the right hand rule.

The component of ${\vec {A}}$ parallel to ${\vec {B}}$ is given by

$\left({\frac {{\vec {B}}\cdot {\vec {A}}}{{\vec {B}}\cdot {\vec {B}}}}\right){\vec {B}}$

and the perpendicular component of ${\vec {A}}$ is given by

$\left({\frac {{\vec {B}}\times {\vec {A}}}{{\vec {B}}\cdot {\vec {B}}}}\right)\times {\vec {B}}$.

This leads to some interesting properties involving combinations of the products, such as

$({\vec {B}}\cdot {\vec {B}}){\vec {A}}=({\vec {A}}\cdot {\vec {B}}){\vec {B}}+({\vec {B}}\times {\vec {A}})\times {\vec {B}}$,

${\vec {A}}\times ({\vec {B}}\times {\vec {C}})=({\vec {A}}\cdot {\vec {C}}){\vec {B}}-({\vec {A}}\cdot {\vec {B}}){\vec {C}}$, and

$({\vec {A}}\times {\vec {B}})\cdot ({\vec {C}}\times {\vec {D}})=({\vec {A}}\cdot {\vec {C}})({\vec {B}}\cdot {\vec {D}})-({\vec {A}}\cdot {\vec {D}})({\vec {B}}\cdot {\vec {C}})$.

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