矢量的加法

Adding Vectors

Revising the definition of a vector

Yes! There are three ways to show a vector, using $\hat i$ and $\hat j$ unit vectors, using column vectors, and the graphical method (drawing it out on the co-ordinate plane)

Adding vectors, graphically

When we added vectors graphically, we use a method called the tip-to-tail method.

In the tip to tail method, when you have two vectors, you put the tail of one vector to the head of the other vector. Then, the sum of the two vectors is just the vector from the unconnected tail to the unconnected head.

This is what it looks like graphically:

Pretty easy, right?

Adding vectors in $\hat i$ and $\hat j$ unit vectors

If you got here then you already understand vector addition in $\hat i$ and $\hat j$ form! All you do is "combine like terms" like you do in algebra!

Adding column vectors

Remember that $3\hat i+4\hat j$ is equivalent to $\begin{pmatrix}3\\4\end{pmatrix}$. Therefore adding column vectors can be defined by:

$$\begin{pmatrix}a_1\\a_2\end{pmatrix}+\begin{pmatrix}b_1\\b_2\end{pmatrix}=\begin{pmatrix}a_1+b_1\\a_2+b_2\end{pmatrix}$$

Now that you know the general idea of what a vector is, let us explore their basic properties. First, let us delve into adding 2 dimensional vectors.

As you already know, 2D vectors are an arrow from one point to another on a coordinate plane. For example:

Now, we must determine what types of vectors we are adding in two dimensions. Usually, vectors are specified as a single point, for example $(6,7)$. This means that the vector is drawn with the 'tail', or 'initial point' at the origin $(0,0)$, and the terminal point, or head, is at $(6,7)$.

Now that the terminology is more clear, let us delve into adding vectors in 2 dimensions. If we wanted to add, for example, $(3,8)$ and $(7,5)$, our sum would be the vector $(3+7,8+5)=(10,13)$.

In general, if we have a sum of two vectors in two dimensions $(a,b)$+$(c,d)$, the sum is $(a+c,b+d)$.

Next, we must figure out why the above is true by drawing parallelograms on the coordinate plane, and translating the vectors.