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# Scalar and Vectors

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Mathematicians and scientists call a quantity which depends on direction a vector quantity, and a quantity which does not depend on direction is called a scalar quantity.
Vector quantities have two characteristics, a magnitude and a direction. Scalar quantities have only a magnitude. When comparing two vector quantities of the same type, you have to compare both the magnitude and the direction. For scalars, you only have to compare the magnitude. When doing any mathematical operation on a vector quantity (like adding, subtracting, multiplying...) you have to consider both the magnitude and the direction. This makes dealing with vector quantities a little more complicated than scalars.
On the slide we list some of the physical quantities discussed in the Beginner's Guide to Propulsion and group them into either vector or scalar quantities. Of particular interest, the forces, which operate on a flying aircraft, the weight, thrust, and aerodynamic forces, are all vector quantities. The resulting motion of the aircraft in terms of displacement, velocity, and acceleration are also vector quantities. These quantities can be determined by application of Newton's laws for vectors. The scalar quantities include most of the thermodynamic state variables involved with the propulsion system, such as the density, pressure, and temperature of the propellants. The energy, work, and entropy associated with the engines are also scalar quantities. There are some quantities, like speed, which have very special definitions for scientists. By definition, speed is the scalar magnitude of a velocity vector. A car going down the road has a speed of 50 mph. Its velocity is 50 mph in the northeast direction. It can get very confusing when the terms are used interchangeably!
While Newton's laws describe the resulting motion of a solid, there are special equations which describe the motion of fluids, gases and liquids, through the propulsion system. For any physical system, the mass, momentum, and energy of the system must be conserved. Mass and energy are scalar quantities, while momentum is a vector quantity. This results in a coupled set of equations, called the Navier-Stokes equations, which describe how fluids behave when subjected to external forces. These equations are the fluid equivalent of Newton's laws of motion and are very difficult to solve and understand. A simplified version of the equations called the Euler equations can be solved for some fluids problems.

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