Why learn about scalar multiplication and vector subtraction?

Many of the fundamental equations in physics are scalar multiplication equations, meaning that we create a new vector by multiplying an initial vector by a scalar. For example, the equation [latex]\vec{B}=c\vec{A}[/latex] is a scalar multiplication equation telling us that we get the vector [latex]\vec{B}[/latex] by multiplying the vector[latex]\vec{A}[/latex] by a scalar [latex]c[/latex]. There are numerous definitions of physical quantities that are vectors that have this form, such as the definition for momentum [latex]\vec{p}=m\vec{v}[/latex] or the gravitational force [latex]\vec{F}_g=m\vec{g}[/latex]. To understand what these definitions are trying to tell us requires knowing how to interpret a scalar multiplication equation.

Often, once we have defined a vector quantity, we will be interested in how it changes. Once I know that the momentum of a particle is given by [latex]\vec{p}=m\vec{v}[/latex], I might want to keep track of how the particles momentum changes. The change in a quantity is always given by the difference between its final value and its initial value. In the case of the momentum of a particle, [latex]\Delta\vec{p}=\vec{p}_f-\vec{p}_i[/latex]. To calculate the change in momentum (or the change in any vector quantity for that matter), I need to work a vector subtraction problem. Given that much of what we will do in this course is keep track of physical quantities as they change, we need to be as proficient with vector subtraction as we are with vector addition.