The \(n\)-dimensional space (or \(n\)-space, in short), denoted by \(\R^n\text{,}\) consists of \(n\)-tuples of real numbers. We call \(\R\) the line, \(\R^2\) the plane, and \(\R^3\) the space.
We usually depict the axes of \(\R^3\) according to the right-hand rule. If you curl the fingers of your right hand from the positive \(x\)-axis (direction of \(\ui\)) toward the positive \(y\)-axis (direction of \(\uj\)), your thumb points in the direction of the positive \(z\)-axis (direction of \(\uk\)). A coordinate system that does not satisfy this is called left-handed.
Elements of \(\R^n\) play a dual role. They are points if \(\R^n\) is regarded as a set, and vectors if \(\R^n\) is regarded as a vector space. In the latter context, real numbers are called scalars. The \(i\)-th component (\(1 \le i \le n\)) of \(\vv = \gv{v_1,\ldots, v_n}\) is \(v_i\text{.}\) The zero vector , denoted by \(\vz\text{,}\) is the vector with all zero components. The position vector of a point \(\vp\) is the same tuple of real numbers regarded as a vector.
In \(\R^n\text{,}\) the vectors \(\mathbf{e}_1 = \gv{1,0,\ldots,0}\text{,}\)\(\mathbf{e}_2 = \gv{0,1,0,\ldots,0}\text{,}\)\(\ldots\text{,}\) and \(\mathbf{e}_n = \gv{0,\ldots,0,1}\) are called the standard basis vectors .
In the special case of \(\R^3\text{,}\) the standard basis vectors \(\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\) are often denoted by \(\ui, \uj, \uk\) respectively. Thus a vector \(\vv = \gv{x,y,z}\) can be written as \(\vv = x\ui + y\uj + z\uk\text{.}\)
Multiplying a vector by a scalar \(\lambda\) changes its length by a factor of \(|\lambda|\text{.}\) Multiplying a vector by \(-1\) reverses its direction.
