Presentation: vector operations

"Physics majors, architects, engineers, lend me your ears, I have come not to introduce vectors..."

"...but to combine them."

Addition and subtraction operations can be done with any number of vectors, but we'll demonstrate on just two vectors at a time, and this discussion can be extended to any number of consecutive operations for larger numbers of vectors.

First we'll discuss the translation property, which allows vectors to be added using a tail-to-tip method. We'll later see that like scalars (numbers), vectors add commutatively.

Recall that a vector is defined by its magnitude (length, a positive-definite quantity) and direction (by convention measured counterclockwise from the +x axis). Translating a vector (that is, moving it around while maintaining its magnitude and direction) in no way changes it. Since in physics, "Everything not forbidden is compulsory" (Murray Gell-Man, citing T.H. White, The Once and Future King, p. 121, Ace, 1996), then translate we must.

Translating vectors such that the tail of a consecutive vector is attached to the tip of the previous vector suggests a path, starting from the first vector through the second vector. The resulting equivalent "shortcut" vector is then the resultant of adding these two vectors.

Let's instead translate these vectors such that their order is switched. As before, the resulting equivalent "shortcut" vector is the resultant of adding these two vectors.

Notice that the resultants of adding these two vectors does not depend on the order. This means that like scalars, the order of addition does not matter, and thus vector addition is commutative.

Let's now do vector subtraction, which builds on our discussion of vector addition.

The twist here is that we can interpret subtraction of a vector as the addition of a negative vector, which results in switching its direction (while keeping the same magnitude).

Now that the subtracted vector is turned around, then after translation, vector subtraction is merely the tail-to-tip vector addition of the first vector to the reversed (subtracted) vector.

Let's see if the order of subtraction is important. We'll reverse the direction of the other vector this time.

Now that the subtracted vector is turned around, then after translation, vector subtraction is still the tail-to-tip vector addition of the first vector to the reversed (subtracted) vector.

Notice that the resultants of subtracting these two vectors does depend on the order. This means that like scalars, the order of subtraction does matter, and thus vector subtraction is non-commutative.

Flashback: trigonometry! Boring, yes, but important.

Briefly, given the magnitude and direction of a vector, the cosine and sine trigonometric functions allow the x- and y- components of a vector to be calculated.

Conversely, given the x- and y- components of a vector, the Pythagorean theorem and the inverse tangent trigonometric function allow the magnitude and direction of a vector to be calculated.

Note that there is a fair amount of care to be taken in determining the right ± signs and angles here, as not all vectors are conveniently going to be located in the first quadrant. Caveat geometer.

But I repeat myself.