Vectors are often used in physics to represent quantities like force and velocity when we need to specify both the magnitude and direction of those quantities. Usually, we represent a vector visually by an arrow. In the applet below, vectors in [math]\mathbb{R}^2[/math] and [math]\mathbb{R}^3[/math] are shown. [br][br][b]Remark[/b]: There is a very special vector that does not have any direction. What is it?

Usually, vectors are drawn as arrows pointing out from the origin (the point where all axes intersect). On a coordinate space (either 2D or 3D), the coordinates of the point at the arrowhead uniquely determine the vector. Therefore, for the vectors v in [math]\mathbb{R}^2[/math] and u in [math]\mathbb{R}^3[/math], we can express them mathematically as follows:[br][br][math]v = \begin{pmatrix} v_x \\ v_y \end{pmatrix} [/math] and [math]u = \begin{pmatrix} u_x \\ u_y \\ u_z \end{pmatrix} [/math][br][br]where [math]\left(v_x,v_y\right)[/math] and [math]\left(u_x,u_y,u_z\right)[/math] are points at the arrowhead of v and u respectively. The above notations for the vectors are called [b]column vectors[/b]. They are actually two specific kinds of [b]matrices[/b]: v and u are [b]2 x 1 matrix[/b] and [b]3 x 1 matrix[/b] respectively.[br][br](You can freely drag the points on the arrowheads of both vectors in the applet above to change the length and the direction of the vectors.)[br][br]Moreover, two vectors are regarded as equal if they have the same length and direction. Hence, sometimes we may shift the vector to another position if necessary, as long as the length and direction remain unchanged.[br][br](You can freely drag the green vectors in the applet above to any position you like and they are considered as the same vectors as the black ones pointing out from the origin.)[br]

As mentioned before, vectors are uniquely determined by the points at their arrowheads when pointing from the origin. Therefore, sometimes it is more convenient to regard vectors as points, especially when we consider not only one, but a set containing many vectors. You will see that it is easier to visualize a set of vectors as a set of points in [math]\mathbb{R}^2[/math] and [math]\mathbb{R}^3[/math].[br][br]In short, vectors and points in [math]\mathbb{R}^2[/math] and [math]\mathbb{R}^3[/math] can be used interchangeably.[br]

We can define vectors in [math] \mathbb{R}^n [/math], where n is any natural number, in a similar way. Obviously, any such vector can be uniquely determined by the n coordinates of its "arrowhead" when the vector is pointing out from the origin. Those n coordinates can be combined to form a column vector, which is in fact a [b]n x 1 matrix[/b] like this:[br][br][math]w=\begin{pmatrix}x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}[/math][br][br]where [math](x_1,x_2, \ldots, x_n)[/math] are the coordinates of the point at the arrowhead of the vector w pointing from the origin in [math]\mathbb{R}^n[/math].[br][br]