Question

How can you geometrically show that two vectors in 2-space or 3-space are equivalent?

Answer 1

If a vector $\mathbf{v}$ in 2 -space or 3 -space is positioned with its initial point at the origin of a rectangular coordinate system, then the vector is completely determined by the coordinates of its terminal point. We call these coordinates the components of $\mathbf{v}$ relative to the coordinate system. We will write $\mathbf{v}=\left(v_{1}, v_{2}\right)$ to denote a vector $\mathbf{v}$ in 2 -space with components $\left(v_{1}, v_{2}\right)$, and $\mathbf{v}=\left(v_{1}, v_{2}, v_{3}\right)$ to denote a vector $\mathbf{v}$ in 3 -space with components $\left(v_{1}, v_{2}, v_{3}\right)$. It should be evident geometrically that two vectors in 2 -space or 3 -space are equivalent if and only if they have the same terminal point when their initial points are at the origin. Algebraically, this means that two vectors are equivalent if and only if their corresponding components are equal. Thus, for example, the vectors $$ \mathbf{v}=\left(v_{1}, v_{2}, v_{3}\right) \quad \text { and } \mathbf{w}=\left(w_{1}, w_{2}, w_{3}\right) $$ in 3 -space are equivalent if and only if $$ v_{1}=w_{1}, \quad v_{2}=w_{2}, \quad v_{3}=w_{3} $$