There are two main operations on vectors. The first one is the addition of two vectors. Let's do the following in the applet below:[br][br][list=1][*]Construct two vectors u and v in [math] \mathbb{R}^2 [/math] using the vector tool [icon]/images/ggb/toolbar/mode_vector.png[/icon].[/*][*]Drag u to the origin i.e. the tail of u is at the origin.[/*][*]Drag v to the arrowhead of u.[/*][*]The vector u + v is defined as the vector pointing from the origin to the arrowhead of v.[br][/*][/list]
Alternatively, you can regard the vector u + v as the "diagonal vector" of the parallelogram formed by the two vectors u and v pointing out from the origin. For the physics viewpoint, this definition of addition is quite natural. You can imagine two forces represented by u and v act on a mass at the origin. The resultant force is exactly u + v.
Draw three vectors u, v, and w in the applet. What can you say about the vectors u + v, v + u, (u + v) + w, and u + (v + w) ?
You can see that u + v = v + u (vector addition is commutative) and (u + v) + w = u + (v + u) (vector addition is associative).
How are the column vectors u and v related to the column vector u + v ? Explain your answer briefly.
The column vector u + v can be obtained by adding the corresponding entries of the column vectors u and v i.e. [math]\begin{pmatrix}u_x\\u_y\end{pmatrix}+\begin{pmatrix}v_x\\v_y\end{pmatrix}=\begin{pmatrix}u_x+v_x\\u_y+v_y\end{pmatrix}[/math].
The definition of addition for higher-dimensional vectors is analogous to the one for vectors in [math] \mathbb{R}^2 [/math] or [math] \mathbb{R}^3 [/math]. It is harder to visualize the addition procedure in higher-dimensional space. Nonetheless, we can define the vector addition by adding the corresponding column vectors in a similar way.[br][br]The following are the random examples of the vector addition of two n-dimensional vectors (expressed as column vectors).[br]