Intro Physics: Vectors

Sections

1. Introduction
2. Vector Components
3. Scalar Multiplication of a Vector and Vector Addition
4. Vector Transformation
5. The Dot Product
6. Rotational Kinematics
7. The Cross Product

---

Introduction

---

In loose physics jargon, vectors are dynamical variables (AKA state variables) with a magnitude and direction.

A more specific definition defines vectors by their coordinate transformation behavior.

Both definition are commonly used and context usually decides which is meant.

Typically, one visualizes a vector as a line aligned with the vector with an arrowhead that gives the sense direction of the the vector : i.e. an arrow symbol: e.g., ← → ↑ ↓ .

In text, one can symbolize/display a vector in several ways:

1. By boldface character: e.g., V

2. By an arrow symbol over a character: just imagine it.

3. In an old-fashioned way favored by yours truly, by a squiggle underneath: just imagine it.

4. In HTML, yours truly uses \vec before a letter: e.g., \vec V. Just accept it.

5. In vector component form as an ordered pair for 2-dimensional cases and as an ordered ordered triple for 3-dimensional cases: e.g., (x,y), (x,y,z).

6. By unit vectors. A unit vector is symbol with a circumflex (AKA hat) on it: e.g., â. But in HTML, yours truly uses \hat before a letter: e.g., \hat V. Just accept it.

7. By a representative general vector component: V_i is the ith vector component and stands for vector V.

   This is compact form is favored by yours truly since it really allows one understand vectors and do vector proofs super-easily though rather abstractly.

Since all symbolizations/displays are used, know them all.

---

Vector Components

---

Vectors must always be specified relative to coordinate system.

A natural way is by length (i.e., magnitude) and angles.

1. In 2-dimensional cases, you can use a length and an angle.
2. In 3-dimensional cases, ou can use a length and two angles.

The length-angle method seems very natural since the length is coordinate-system invariant.

The angles are always dependent on the coordinate system.

However, the length-angle method is intractable for computations and proofs because the length and angles have to be treated differently. They have different natures: i.e., different physical dimensions.

The less pictural, but tractable method is by vector components.

The components of a vector are the vector projections of the the vector along the coordinate axes.

The vector projection of a vector is the length of the vector times the cosine of the angle makes with a coordinate axis: e.g.,

vector \vec a has component a_i = a*cos(θ_i), where a is the length of the vector and &theta_i is the angle between the vector and axis i.

The length of a vector \vec a follows from the Pythagorean theorem. Behold:

   a**2 = a_i*a_i 

where a without adornment is understood to be length, i is coordinate index, and there is a summation on the repeated index i.
Summation on a repeated index is the Einstein summation rule. It's a great compactification since get rid of klutzy summation signs ∑. However, sometimes you have to turn it off. Usually context tells you whether it's being used or not.

---

Scalar Multiplication of a Vector and Vector Addition

---

---

Vector Transformation

---

Let T be a transformation matrix which we symbolize by a representative element T_ij (which is the matrix element in the ith row and jth column).

We can a vector a_i into any vector b_i by using the appropriate T_ij.

Behold:

      b_j = T_ij * a_j

where there is summation on the repeated index j.

Now changing one vector into another is interesting in itself if the vector represents a dynamical variable and there is physical process causing the transformation.

But at the moment, let's consider vector transformation that is a length conserving: this kind is called an orthogonal transformation.

Orthogonal transformations involve vector in Euclidean space and in intro physics we usually only treat 1-, 2-, and 3-dimensional Euclidean space.

The generalization of orthogonal transformation to treat Hilbert space and complex numbers is called unitary transformation.

So an orthogonal transformation is a unitary transformation, but NOT vice versa in general.

Let a_i' be the orthogonal transformation of a_i. Behold:

   a_i' = U_ij * a_j ,

where U_ij is the orthogonal transformation.

What condition does a orthogonal transformation obey. Well a'**2 = a_i'*a_i' = a_i*a_i = a**2 since length is conserved.

Thus,

    a'**2 = a_i'*a_i' = U_ij*U_ik*a_j*a_k = a**2  .

We can satisfy equation if

   U_ij*U_ik = δ_ij  ,

where δ_ij is the Kronecker delta which has the property:

  
   δ_ij = 0 if i≠j
              = 1 if i=j  . 

The Kronecker delta rank 2 tensor---a tensor is a generalization of a vector.

Now let U**T be the transpose matrix of matrix U. We then see that

   U_ij*U_ik = U**T_ji*U_ik = δ_ij  ,

which implies

   U**T = U**(-1) , 

U**2 is the inverse matrix. The equation U**T = U**(-1) is, in fact, the defining equation of an orthogonal transformation.

What is an orthogonal transformation?

It can be considered a rotation and/or inversion of vector.

But it can also be considered a rotation and/or inversion of of the coordinate system used to describe the vector.

In this latter case, nothing changes about the vector physically, only our description of it changes.

Note that considered as coordinate-system transformation, the coordinate axes are transformed oppositely to the way a vector is transformed if the orthogonal transformation is thought of as transfroming the vector.

This simple point can lead to lots of confusion.

Let's see if we can clarify once for all.

Say we have orthogonal transformation U.

Applied to vector a, we NEW vector

    a_i' = U_ij*a_j
   

Now we consider both a and a' in a coordinate system with unit vectors e_i. The unit vector point in the directions of the coordinate axes.

This means

a_i = a

---

The Levi-Civita Symbol

---

Before going onto the cross product in section The Cross Product, a digression on the (3-dimensional) Levi-Civita symbol is needed.

The Levi-Civita symbol (pronunciation levi-chivita: see Forvo: Levi-Civita) is defined

  ε_ijk =  1 if ijk are cyclic:  e.g., 123 
                  -1 if ijk are anticyclic:  e.g., 213 
                   0 if ijk are not distinct:  e.g., 113

where ijk are indices of the Cartesian coordinates: 1 stands for x, 2 stands for y, 3 stands for z.

The Levi-Civita symbol is defined to be coordinate-system---which means it is NOT a tensor---it is a tensor density.

Levi-Civita symbol has at least two important uses: 1) it can be used to determine the determinant, 2) it is used to write the cross product in component.

The Levi-Civita symbol cannot be expanded in terms of Kronecker deltas---which must be true since yours truly has never seen it done and can't figure out how it could be done.

But there are some very useful identities involving the Levi-Civita symbol and the Kronecker delta:

1. The first identity may not be so important in itself, but it allows one to prove other identities easily. Let's just find the identity in its proof.

   By inspection,

            ε_ijk*ε_lmn =  1 if ijk and lmn have the same cyclicity
                                        = -1 if ijk and lmn have the different cyclicity
                                        =  0 if ijk or lmn are not distinct.
          

   Using the Einstein summation rule and the result of the last result, we find

            ε_ijk*ε_lmn = ε_i'j'k'*ε_lmn*δ_ii'*δ_jj'*δ_kk'
    
                =   δ_il*δ_jm*δ_kn
                  + δ_in*δ_jl*δ_km
                  + δ_im*δ_jn*δ_kl
                  - δ_im*δ_jl*δ_kn
                  - δ_in*δ_jm*δ_kl
                  - δ_il*δ_jn*δ_km   .
          

   which is the result identity: QED.

   Note we used that ε_i'jk*δ_ii' = ε_ijk, etc. and as we cycled through i'j'k' summations we used the now 2nd to last result.

   The proof seems valid to me if lmn are NOT distinct. It never uses that they are distinct. ????? yes I do, drat, I assume that lmn are distinct so that i'j'k' can run over them????

2. The second identity gets a lot of use in proving cross-product identities.

   Starting from the first identity with l set to i and a summation on i, we get

            ε_ijk*ε_imn 
   
                =   δ_ii*δ_jm*δ_kn
                  + δ_in*δ_ji*δ_km
                  + δ_im*δ_jn*δ_ki
                  - δ_im*δ_ji*δ_kn
                  - δ_in*δ_jm*δ_ki
                  - δ_ii*δ_jn*δ_km
   
                = 3*δ_jm*δ_kn + δ_jn*δ_km
                                          + δ_jn*δ_km
                  - δ_jm*δ_kn - δ_jm*δ_kn
                 -3*δ_jn*δ_km
   
                = δ_jm*δ_kn - δ_jn*δ_km   .
            

   So the identity with indices relabeled to conventional ones is

    
            ε_ijk*ε_ilm = δ_jl*δ_km - δ_jm*δ_kl   .
            

3. The third identity follows from the second with l set to j and j summed on:

           ε_ijk*ε_ijm = δ_jj*δ_km - δ_jm*δ_kj 
                                       = 3*delta;_km - delta;_km
                                       = 2*delta;_km
            

   So the identity with indices relabeled to conventional ones is

            ε_ijk*ε_ijl = 2*delta;_kl .
            

4. The fourth identity follows from the third with l set to k and k summed on:

           ε_ijk*ε_ijk = 6  .
           

   This last result can be proven directly by inspection.
As last topic, without proof, we give the following result

     det|U|*ε_ijk = ε_i'j'k'*U_ii'*U_jj'*U_kk'  ,
   

where U is an orthogonal transformation and det|U| is the determinant of U.

Recalling that det|U| = ±1 with the upper case for the no inversion and the lower case for with inversion we find that

     ε_ijk = det|U| U_ii'*U_jj'*U_kk'*ε_i'j'k'
   

which shows how the Levi-Civita symbol transforms under an orthogonal transformation.

If the Levi-Civita symbol were a third-rank tensor, there would be no factor of det|U|.

---

The Cross Product

---

There is a third kind of vector multiplication beyond scalar multiplication of a vector dot product: the cross product.

1. Definition the Cross Product:

   The definition is

         \vec c = \vec a x \vec b = ab*sin(θ)\hat c
   

   where θ is the angle between \vec a and \vec b (with their tails joined figuratively speaking) and \hat c is unit vector perpendicular to \vec a and \vec b: i.e., pointing out of the plane defined by \vec a and \vec b.

   What is the sense direction of \hat c?

   It is determined a right-hand rule: sweep the fingers of your right hand from the first vector in the cross product toward the second, and your right-hand points, more or less, in the sense direction of \hat c.

   The right-hand rule is, in fact, an arbitrary convention in all physics it seems, except for parity violation effects (see Wikipedia: Pseudovector: The right-hand rule). It was so chosen because right-handers are in charge of the universe---sorry southpaw. You have to maintain the convention to make everything work out right.

   We explain why we need a right-hand rule in the next section.

   Some simple cross-product results:

    
          \vec a x \vec b = ab*sin(θ =  0°)\hat c   = 0
          \vec a x \vec b = ab*sin(θ = 90°)\hat c   = ab\hat c 
          \vec a x \vec b = ab*sin(θ =180°)\had c   = 0 
          \vec b x \vec a = - ab*sin(θ = 80°)\hat c = -\vec a x \vec b   ,
      

   where the last result shows that the cross product has anticommutativity.

2. The Cross Product is an Axial Vector:

   The cross product is, in fact, an axial vector (AKA pseudovector), NOT an ordinary vector. Another way of saying that is that it is NOT a vector, but it acts like vector in many respects.

   Mathematically, an axial vector has a different coordinate-system transformation than an orthogonal transformation. For rotations, it's the same. For inversions, there is an extra multiplication by -1.

   In physics, we almost never have to worry about what happens on inversions.

   As an example, a physics-relevant cross-product axial vector is angular momentum.

   Angular momentum for a point particle is defined

    
           \vec L = \vec r x \vec p = \vec r x m\vec v  ,
      

   where \vec r is the displacement vector from an origin, \vec p is the momentum (AKA linear momentum), and m is the mass.

   You will note that angular momentum is origin dependent.

   You will also note that \vec L points along the rotational axis of the point particle in the sense direction right-hand rule.