Basic intro to vectors, addition and multiplication by a scalar.

Thinking about vectors

          Vectors, for the purposes of elementary physics, is a quantity which has two separate pieces of information embedded in it: magnitude (sometimes called the norm) and direction. At least this is the interpretation which we will use. Vectors are often depicted as arrows, where the arrow's length typically corresponds with the magnitude, and its orientation corresponds with direction. Vectors can be depicted in one, two, or three dimensions, and can even live in n dimensions greater than three, but then it becomes impossible to visualise as a simple arrow in space. For the purposes of using vectors for physical phenomena such as electromagnetism, we shall constrain ourselves to no more than three dimensional space. 

How is magnitude and direction defined?

          To describe a vector fully the direction and magnitude has to be specified. There are many ways of doing this, and they are all correct so long as there is absolutely no ambiguity over what the magnitude and direction of the vector being described. The methods of describing it will be described later on in this article.

A Useful property of vectors

        

          Due to the way that we have defined a vector as being a quantity of only magnitude and direction, a useful property is implied when we imagine vectors as arrows in space. 

      I like to think of a vector as a set of instructions for movement: "Walk x metres in this direction." Then, imagine asking it, "Ok, where do I start from?" Then the vector will reply: "I don't care."

          The vector only gives information about the direction and magnitude of movement. Hence, it doesn't matter where the initial point is, and by that logic that implies that  two vectors which have the same magnitude and direction but placed at the point (2,2) and the other placed at (-1000,23.4533234) are equivalent. 

          Since it doesn't matter where you root your vector, in mathematics we tend to place the tail at the origin because it makes calculations simple.

Two elementary operations on vectors

        There are two fundamental operations which one can perform on a vector. It is vector addition, and scalar multiplication. There are more of course, but these two are the springboard. 

  Vector addition

          Suppose that we take a really simple space where we can perform our vector operations in, the one-dimensional number line. And suppose, vector v tells you to move 6 meters to the right, and vector w tells you to move 3 meters to the left. Now, imagine that we were interested in the combined effect of v and w. What does doing v first and then doing w second effectively amount to? In other words, what does v+w look like?

        We can display v and w as two separate arrows. Here they are:

vector v

vector w

        Now, try to imagine a graphical way to represent this combined effect that v and w has. 

Hint: use the previously discussed fact that you can slide any of the two vectors along anywhere that you desire along the number line without changing its identity.

        It turns out that if we take w and place the tail of the arrow onto the tip of v, then the tip of w lies exactly on top of the answer. Then, we can even further extend this idea, and define a new vector v+w to start at the point where we started, and end at the point where we ended. This new vector tells you straight away where you will end up without having to do v and w seperately.

"Adding" the vectors

v+w

 

      Now, importantly, this is the geometric equivalent of saying 3=(5)+(-2). And, just like how we can equivalently say that 3=(-2)+(5), it is the case that v+w=w+v.

     This can also be done in two dimensions and three dimensions. As a general rule, to find v+w, bring the tail of w so it touches the tip of v, and drawing an arrow from the tail of v to the tip of the newly placed w gives you the vector v+w. This can be done in the opposite order too.

Vector v and w rooted at the origin

Sliding or "translating" one of the vectors is allowed

The vector v+w starts at what I like to call the "free ends": the tail of the vector that wasn't moved, and the tip of the vector
that was moved.

Scalar multiplication

           We used the idea of addition from arithmetic to explain vector addition, and show that they are fundamentally the same idea. What about multiplication? 

           Well, suppose that we take vector v which tells you how far you have walked along the number line. Then, what would the vector 2v look like? That is easy. If you walked twice as far, the arrow representing the vector will be twice as long. In general, the magnitude of the vector cv is simply c times the magnitude of v. 

The Cartesian Starter Pack: Basis Vectors.

          Now, returning to the xy plane, there really are two vectors that you will ever need. A vector, which I will from now on use interchangeably with "arrow" which points horizontally and is of magnitude one, and another vector which points vertically and is of magnitude one as well.

We give these vectors names: i-hat for the former (which I will denote as simply i, don't confuse it with i from complex numbers), and j-hat for the latter (which I will also denote as j). 

        

 Take the i-hat unit vector, and scale it to the vector xi. Next, take the j-hat unit vector, and scale it to yj. 

I used x0 and y0 instead of just saying x and y to emphasise that these are arbitrary but particular points on the plane. There is no effective difference though, that is my choice of notation.

        Then, if you bring the tail of the vector yj to the tip of xi, it will touch the point (x,y). But this is vector addition, so this means that the vector v really is the same thing as saying xi+yj. 

That helpfully breaks our vector up into horizontal and vertical components, which is a very common practice in physics and mathematics- practice doing this!

           Hence, a vector which is rooted at the origin and represents a point (x,y) is given by v=xi+yj.

This vector describes a unique position on the x,y plane, so we give it a special name: the position vector, which you met in my previous blog.

           This can be extended to three dimensions, where the basis vectors are i,j, and k for the x,y, and z axis respectively.

Column vector notation

          Another popular way of denoting a vector v=xi+yj is also by writing the x and y components in a column vector form, shown blow.

V can be therefore written in terms of the i and j hat unit vectors, or in column vector notation.  A vector which is
rooted in the origin is often called a position vector, and the letter used instead of v often is r. 

Algebraic and Analytical vector addition and vector-scalar multiplication

  Let's now try to add two vectors v and w with x components x1 and x2 respectively, and y components y1 and y2 respectively. From combining like terms, we can algebraically prove that the sum of two vectors is the same as the sum of the corresponding components:

Similarly, we can also show that  the product of a vector and a scalar c is equivalent to the product of the x-component and c, the y-component and c, and so on:

           In column vector notation form: