Components of Vector between two Points

Theorem

Let $A, B$ be points in the Euclidean space $\R^n$.

Let their Cartesian coordinates be given by:

$\ds A$

$=$

$\ds \tuple {a_1 , a_2, \ldots, a_n }$

$\ds B$

$=$

$\ds \tuple {b_1 , b_2, \ldots, b_n }$

Let $\vec {AB}$ be the vector quantity that represents the directed line segment with startpoint $A$ and endpoint $B$.

Then the vector components of $\vec {AB}$ are:

$\vec {AB} = \tuple {b_1 - a_1, b_2 - a_2, \ldots, b_n - a_n}$

Let $O$ denote the origin of $\R^n$.

Let $\vec {OA}$ and $\vec {OB}$ be the positions vectors of $A$ and $B$.

By definition of positions vectors, their components are:

$\ds \vec {OA}$

$=$

$\ds \tuple {a_1 , a_2, \ldots, a_n }$

$\ds \vec {OB}$

$=$

$\ds \tuple {b_1 , b_2, \ldots, b_n }$

for some $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \R$.

Let the components of $\vec {AB}$ be denoted as:

$\vec {AB} := \tuple {v_1 , v_2, \ldots, v_n}$

By the definition of Triangle Law of Vector Addition, it follows that $\vec {OB}$ can be represented as:

$\vec {OB} = \vec{OA} + \vec{AB}$

$\tuple {b_1 , b_2, \ldots, b_n} = \tuple {a_1 , a_2, \ldots, a_n} + \tuple {v_1 , v_2, \ldots, v_n}$

By the definition of vector addition of components, it follows that:

$\tuple {b_1 - a_1, b_2 - a_2, \ldots, b_n - a_n} = \tuple {v_1 , v_2, \ldots, v_n}$

Hence the result.

$\blacksquare$