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Calculating using vectors: Velocity^2 vs. DOT(Velocity,Velocity)

Same caution needs to be used when implementing in the calculator equations containing the square of a vector field: what the equation is exactly indicating can indeed be not so clear from the mathematical formulation.


Let's assume for example that you want to calculate the the dynamic pressure as a scalar:

dynamic_pressure = 0.5 * Density * Velocity^2  (eqn #1)

where velocity is a vector [vx,vy,vz] and Density is a scalar.

If you simply use the expression Velocity^2 in the calculator, you are actually using a vector with components [vx*vx, vy*vy, vz*vz] and with magnitude = sqrt(vx^4+vy^4+vz^4), and therefore the expression above in equation #1 would become:

dynamic_pressure = 0.5*Density*sqrt(vx^4+vy^4+vz^4)

and return a vector field instead of a scalar. The correct way to calculate this quantity is to use the expression:

dynamic_pressure = 0.5*Density*DOT(Velocity,Velocity)  (eqn #2)

which will return a scalar of magnitude 0.5*Density*(vx^2+vy^2+vz^2)

The difference between the vector magnitude in equation #1 and the scalar in equation #2 becomes more pronounced as velocity gets larger.  

On a more general note, please keep in mind:

Scalars that are equivalent to each other

  •  DOT(velocity,velocity)
  •  RMS(velocity) * RMS(velocity)
  • (Velocity[x]*Velocity[x] + Velocity[y]*Velocity[y]+Velocity[z]*Velocity[z]) 


Vectors that are equivalent to each other.

  • Velocity^2
  • Velocity * Velocity. They both give you a vector of magnitude sqrt(vx^4+vy^4+vz^4)
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