The Physics
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Opus in profectus

Vector Multiplication

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Practice

practice problem 1

Write something.

solution

Answer it.

practice problem 2

Write something.

solution

Answer it.

practice problem 3

Derive the law of cosines from the dot product.

solution

Begin by defining an arbitrary vector C as the difference between two other vectors A and B, then take the dot product of C with itself.

Let…

C = A − B

Then…

C · C  = (A − B) · (A − B)
C2  = (A · A) − (A · B) − (B · A) + (B · B)
C2  = A2 + B2 − 2AB cosθ
 

practice problem 4

Test the cross product for associativity by determining if this equation is true.

(A × B) × C ≟ A × (B × C)

solution

Behold! A big damn pile of symbols.

We've already shown that…

A × B = (AyBz − AzBy + (AzBx − AxBz + (AxBy − AyBx

Change the symbols around, swapping A with B and B with C.

B × C = (ByCz − BzCy + (BzCx − BxCz + (BxCy − ByCx

Now for the tedious part. Take the first equation and cross it into C.

(A × B) × C  = (AyBz − AzBy × Cx  + (AzBx − AxBz × Cy  + (AxBy − AyBx × Cz  + (AyBz − AzBy × Cx  + (AzBx − AxBz × Cy  + (AxBy − AyBx × Cz  + (AyBz − AzBy × Cx  + (AzBx − AxBz × Cy  + (AxBy − AyBx × Cz 

Eliminate the zero terms. Watch the signs on the other terms.

(A × B) × C  =  (AzBxCy − AxBzCy)    −  (AxByCz − AyBxCz)  
   −  (AyBzCx − AzByCx)    +  (AxByCz − AyBxCz)  
   +  (AyBzCx − AzByCx)    −  (AzBxCy − AxBzCy)  

Then simplify.

(A × B) × C  =  [(AxByCz + AxBzCy)  −  (AyBxCz + AzBxCy)]  
   +  [(AyBxCz + AyBzCx)  −  (AxByCz + AzByCx)]  
   +  [(AzBxCy + AzByCx)  −  (AxBzCy + AyBzCx)]  

Repeat by crossing A into the second equation.

A × (B × C)  = Ax  × (ByCz − BzCy + Ax (BzCx − BxCz × + Ax  × (BxCy − ByCx + Ay  × (ByCz − BzCy + Ay (BzCx − BxCz × + Ay  × (BxCy − ByCx + Az  × (ByCz − BzCy + Az (BzCx − BxCz × + Az  × (BxCy − ByCx

Eliminate the zero terms. Watch the signs on the other terms.

A × (B × C)  =  (AxBzCx − AxBxCz  −  (AxBxCy − AxByCx
   −  (AyByCz − AyBzCy  +  (AyBxCy − AyByCx
   +  (AzByCz − AzBzCy  −  (AzBzCx − AzBxCz

Then simplify.

A × (B × C)  =  [(AyBxCy + AzBxCz)  −  (AyByCx + AzBzCx)]  
   +  [(AxByCx + AzByCz)  −  (AxBxCy + AzBzCy)]  
   +  [(AxBzCx + AyBzCy)  −  (AxBxCz + AyByCz)]  

Are the two products equal or are they not?

(A × B) × C ≟ A × (B × C)

Let's make a direct comparison of the components.

[(AxByCz + AxBzCy) − (AyBxCz + AzBxCy)]   ≟  [(AyBxCy + AzBxCz) − (AyByCx + AzBzCx)] 
 
[(AyBxCz + AyBzCx) − (AxByCz + AzByCx)]   ≟  [(AxByCx + AzByCz) − (AxBxCy + AzBzCy)] 
 
[(AzBxCy + AzByCx) − (AxBzCy + AyBzCx)]   ≟  [(AxBzCx + AyBzCy) − (AxBxCz + AyByCz)] 

I don't see one triplet of subscripts in the same order as any other, therefore the vector cross product is not an associative operation.

(A × B) × C ≠ A × (B × C)

Well now, that wasn't any fun, but fun be damned. This ain't no amusement park. It's a math proof.