FHSST Physics/Vectors/Mathematical Properties

Mathematical Properties of Vectors

Vectors are mathematical objects and we will use them to describe physics in the language of mathematics. However, first we need to understand the mathematical properties of vectors (e.g. how they add and subtract).

We will now use arrows representing displacements to illustrate the properties of vectors. Remember that displacement is just one example of a vector. We could just as well have decided to use forces to illustrate the properties of vectors.

Addition of Vectors

If we define a displacement vector as 2 steps in the forward direction and another as 3 steps in the forward direction then adding them together would mean moving a total of 5 steps in the forward direction. Graphically, this can be seen by first following the first vector two steps forward, and then following the second one three steps forward:

We add the second vector at the end of the first vector, since this is where we now are after the first vector has acted. The vector from the tail of the first vector (the starting point) to the head of the last (the end point) is then the sum of the vectors. This is the tail-to-head method of vector addition.

The order in which you add vectors does not matter. In the example above, if you decided to first go 3 steps forward and then another 2 steps forward, the end result would still be 5 steps forward.

The final answer when adding vectors is called the resultant.

Definition: The resultant of a number of vectors is the single vector whose effect is the same as the individual vectors acting together.

In other words, the individual vectors can be replaced by the resultant — the overall effect is the same. If vectors ${\overrightarrow {a}}$ and ${\overrightarrow {b}}$ have a resultant ${\overrightarrow {R}}$, this can be represented mathematically as, ${\begin{matrix}{\overrightarrow {R}}&=&{\overrightarrow {a}}+{\overrightarrow {b}}.\end{matrix}}$

Let us consider some more examples of vector addition using displacements. The arrows tell you how far to move and in what direction. Arrows to the right correspond to steps forward, while arrows to the left correspond to steps backward. Look at all of the examples below and check them.

Let us test the first one. It says one step forward and then another step forward is the same as an arrow twice as long — two steps forward.

It is possible that you end up back where you started. In this case the net result of what you have done is that you have gone nowhere (your start and end points are at the same place). In this case, your resultant displacement is a vector with length zero units. We use the symbol ${\overrightarrow {0}}$ to denote such a vector:

Check the following examples in the same way. Arrows up the page can be seen as steps left and arrows down the page as steps right.

Try a couple to convince yourself!

It is important to realise that the directions aren't special — forward and backwards or left and right are treated in the same way. The same is true of any set of parallel directions:

In the above examples the separate displacements were parallel to one another. However the same tail-to-head technique of vector addition can be applied to vectors in any direction.

Now you have discovered one use for vectors; describing resultant displacement — how far and in what direction you have travelled after a series of movements.

Although vector addition here has been demonstrated with displacements, all vectors behave in exactly the same way. Thus, if given a number of forces are acting on a body, you can use the same method to determine the resultant force acting on the body. We will return to vector addition in more detail later.

Subtraction of Vectors

What does it mean to subtract a vector? Well this is really simple: if we have 5 apples and we subtract 3 apples, we have only 2 apples left. Now lets work in steps — if we take 5 steps forward, and then subtract 3 steps forward, we are left with only two steps forward:

What have we done? You originally took 5 steps forward but then you took 3 steps back. That backward displacement would be represented by an arrow pointing to the left (backwards) with length 3. The net result of adding these two vectors is 2 steps forward:

Thus, subtracting a vector from another is the same as adding a vector in the opposite direction (i.e. subtracting 3 steps forwards is the same as adding 3 steps backwards).

This suggests that in this problem, arrows to the right are positive, and arrows to the left are negative. More generally, vectors in opposite directions differ in sign (i.e. if we define up as positive, then vectors acting down are negative). Thus, changing the sign of a vector simply reverses its direction:

In mathematical form, subtracting ${\overrightarrow {a}}$ from ${\overrightarrow {b}}$ gives a new vector ${\overrightarrow {c}}$

${\begin{matrix}{\overrightarrow {c}}&=&{\overrightarrow {b}}-{\overrightarrow {a}}\\&=&{\overrightarrow {b}}+(-{\overrightarrow {a}})\end{matrix}}$

This clearly shows that subtracting vector ${\overrightarrow {a}}$ from ${\overrightarrow {b}}$ is the same as adding $(-{\overrightarrow {a}})$ to ${\overrightarrow {b}}$. Look at the following examples of vector subtraction.

Scalar Multiplication

What happens when you multiply a vector by a scalar (an ordinary number)?

Going back to normal multiplication we know that $2\times 2$ is just 2 groups of 2 added together to give 4. We can adopt a similar approach to understand how vector multiplication works.