Division of physical vectors

A physical vector is a physical magnitude with a direction that operates as a mathematical vector. As with all physical quantities, it has units of some kind. Both the magnitude and the direction have units. The directional units are called unit vectors.

The units of a magnitude are what it is relative to, for example: metres per second means the distance traveled in metres relative to a travel time in seconds along a particular direction. If we multiply a velocity vector by a time interval, the result is a position vector. What has changed is the units.

The vector formed by the division of two vectors with the same direction has a magnitude that is the ratio of the two magnitudes. Its direction is the same as the direction of the two vectors. So the division of vector s by vector t is the ratio of their magnitudes times their common unit vector:

s / t = (|s|/|t|) (s/|s|) = s / |t|.

That is, the ratio of two vectors equals the numerator vector divided by the magnitude of the denominator vector. Since time is often the denominator vector in which only the magnitude is needed, the vector nature of time has been hidden.

What about vector division for vectors that might be in different directions? What is the magnitude and direction in that case? Look at the vector division above a little differently:

s / t = (|s| s ) / (|s| |t|)

The denominator above gives us a clue: instead of the product of the magnitudes, their dot product can be used, since

s ∙ t = |s| |t| cos θ,

where θ is the angle between vectors s and t. Then put this dot product into the denominator and leave the numerator the same:

s / t = (|s| s ) / (s ∙ t) = s / (|t| cos θ).

So vector division is the denominator vector divided by the magnitude of the projection of the denominator vector onto the numerator vector. Although the direction of the numerator is the direction of the division, the direction applies to the whole magnitude, not just the numerator.