# Curiosity with the Cartesian Notation of the Vector Cross Product

In my opinion Hibbeler's book on statics (Engineering Mechanics Statics, 12th ed) is one of the most approachable on the subject. On pg.123 he defines the Vector Cross Product in its Cartesian notation as follows; $$\vec{A}\times\vec{B}=(A_yB_z-A_zB_y)\vec{i}-(A_xB_z-A_zB_x)\vec{j}+(A_xB_y-A_yB_x)\vec{k}\\$$ I understand the $\vec{i},\vec{j}\;{and}\;\vec{k}$ terms represent unit vectors which represent the direction of the magnitude of the $\vec{A}$ components designated by $A_x,\;A_y\;{and}\;A_z$, and the $\vec{B}$ components designated by $B_x,\;B_y\;{and}\;B_z$.

My question is, why is this relationship not defined as follows; $$\vec{A}\times\vec{B}=(A_yB_z-A_zB_y)\vec{x}-(A_xB_z-A_zB_x)\vec{y}+(A_xB_y-A_yB_x)\vec{z}\\$$ Replacing the $\vec{i},\vec{j}\;{and}\;\vec{k}$ terms with $\vec{x},\vec{y}\;{and}\;\vec{z}$, so as to be consistent with the notation of the vector components? What is gained by the introduction of this additional representation of these dimensions?

Thank you

• They are still the standard unit vectors, $(1,0,0),(0,1,0),(0,0,1)$ no matter what letters are used to denote them. – coffeemath Feb 21 '15 at 12:11
• @coffeemath Thank you for taking the time to look at this. I understand and agree completely with what you are saying. My question is why they would be different. I remember when I first encountered vector notation, this was a source of some confusion. I see it repeatedly from different sources which makes me think this notation must be adding to the relationship in a way I do not appreciate. – James Izzard Feb 21 '15 at 12:16
• It seems to be related to quaternions, see answer... – coffeemath Feb 21 '15 at 12:32
• He uses the same i,j,k-notation for 3D unit vectors through the book, e.g. the same appears on p. 106. Where did you see him use x,y,z as the unit vectors? As far as I can tell he never introduces quaternions, by the way. So it's just a matter of choice of notation. The i,j,k notation is pretty standard: en.wikipedia.org/wiki/Unit_vector (although by no means the only one found in other texts). – Fizz Feb 21 '15 at 16:25
• @RespawnedFluff To my knowledge he does not use x,y and z to denote unit vectors. However he does use subscript x,y and z to denote components of a vector. As far as I could see it seemed strange use both x and i, y and j, z and k to denote the same three dimensions. Why not choose one and stick with it? – James Izzard Feb 21 '15 at 16:32

## 1 Answer

If we take 3-space as embedded in the four dimensional real quaternions, as those elements having first coordinate $0$, then a typical element is $0+xi+yj+zk$ and there is a multiplication defined on the quaternions by defining $ij=k,\ jk=i,\ ki=j,$ with the other products being $ji=-k,\ kj=-i,\ ik=-j,$ then the cross product corresponds to quaternion multiplication.

Note one also defines $ii=jj=kk=0$ and extends a multiplication linearly.

Added note: The question was about why the specific letters $i,j,k$ are sometimes used for the unit vectors in the definition of cross product. What I'm getting at here is that it may be due to the facts that (1) These letters are almost universally used in describing the quaternions, and (2) the quaternion multiplication on the last three coordinates happens to correspond componentwise to the cross product.

• I must admit I do not currently understand quaternions, and so you leave me somewhat out of my depth. However I gather what you are saying is the cross product is defined as such to promote consistency with other relationships with which the reader may not yet be familiar. Please correct me if I am wrong in this understanding. Thank you. – James Izzard Feb 21 '15 at 12:38
• The cross product $(x,y,z) \times (x',y',z')$ turns out to be in quaternion components $(u,v,w)=ui+vj+wk$ to be the same as multiplying the two quaternion versions of the given starting vectors. For example $xiy'j=(xy')(ij)=xy'k,$ which explains one of the two terms in the last coordinate of the cross product. – coffeemath Feb 21 '15 at 13:06
• @JamesIzzard I just put in an extra part in the above, maybe a bit more about why specifically $i,j,k$ are sometimes chosen for the unit vectors. – coffeemath Feb 21 '15 at 13:29
• I have just noticed the following statement on the wikipedia page: (en.wikipedia.org/wiki/Cross_product) "The cross product can also be described in terms of quaternions, and this is why the letters i, j, k are a convention for the standard basis on R3. The unit vectors i, j, k correspond to "binary" (180 deg) rotations about their respective axes (Altmann, S. L., 1986, Ch. 12), said rotations being represented by "pure" quaternions (zero scalar part) with unit norms". Seems you were absolutely right. – James Izzard Feb 23 '15 at 13:44
• @JamesIzzard Yes, that backs up what I had thought, can't recall where I learned of that approach to cross products. – coffeemath Feb 23 '15 at 13:52