Question

What are standard unit vectors?

Answer 1

When a rectangular coordinate system is introduced in $R^{2}$ or $R^{3}$, the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors. In $R^{2}$ these vectors are denoted by $$ \mathbf{i}=(1,0) \text { and } \mathbf{j}=(0,1) $$ and in $R^{3}$ by $$ \mathbf{i}=(1,0,0), \quad \mathbf{j}=(0,1,0), \quad \text { and } \quad \mathbf{k}=(0,0,1) $$ Every vector $\mathbf{v}=\left(v_{1}, v_{2}\right)$ in $R^{2}$ and every vector $\mathbf{v}=\left(v_{1}, v_{2}, v_{3}\right)$ in $R^{3}$ can be expressed as a linear combination of standard unit vectors by writing $$ \begin{aligned} &\mathbf{v}=\left(v_{1}, v_{2}\right)=v_{1}(1,0)+v_{2}(0,1)=v_{1} \mathbf{i}+v_{2} \mathbf{j} \\ &\mathbf{v}=\left(v_{1}, v_{2}, v_{3}\right)=v_{1}(1,0,0)+v_{2}(0,1,0)+v_{3}(0,0,1)=v_{1} \mathbf{i}+v_{2} \mathbf{j}+v_{3} \mathbf{k} \end{aligned} $$ Moreover, we can generalize these formulas to $R^{n}$ by defining the standard unit vectors in $R^{n}$ to be $$ \mathbf{e}_{1}=(1,0,0, \ldots, 0), \quad \mathbf{e}_{2}=(0,1,0, \ldots, 0), \ldots, \quad \mathbf{e}_{n}=(0,0,0, \ldots, 1) $$ in which case every vector $\mathbf{v}=\left(v_{1}, v_{2}, \ldots, v_{n}\right)$ in $R^{n}$ can be expressed as $$ \mathbf{v}=\left(v_{1}, v_{2}, \ldots, v_{n}\right)=v_{1} \mathbf{e}_{1}+v_{2} \mathbf{e}_{2}+\cdots+v_{n} \mathbf{e}_{n} $$