Definition:Vector Sum

From ProofWiki
Jump to navigation Jump to search

Definition

Let $\mathbf u$ and $\mathbf v$ be vector quantities of the same physical property.


Component Definition

Let $\mathbf u$ and $\mathbf v$ be represented by their components considered to be embedded in a real $n$-space:

\(\displaystyle \mathbf u\) \(=\) \(\displaystyle \tuple {u_1, u_2, \ldots, u_n}\)
\(\displaystyle \mathbf v\) \(=\) \(\displaystyle \tuple {v_1, v_2, \ldots, v_n}\)


Then the (vector) sum of $\mathbf u$ and $\mathbf v$ is defined as:

$\mathbf u + \mathbf v := \tuple {u_1 + v_1, u_2 + v_2, \ldots, u_n + v_n}$


Note that the $+$ on the right hand side is conventional addition of numbers, while the $+$ on the left hand side takes on a different meaning.

The distinction is implied by which operands are involved.


Triangle Law

Let $\mathbf u$ and $\mathbf v$ be represented by arrows embedded in the plane such that:

$\mathbf u$ is represented by $\vec {AB}$
$\mathbf v$ is represented by $\vec {BC}$

that is, so that the initial point of $\mathbf v$ is identified with the terminal point of $\mathbf u$.

Vector-sum-triangle-law.png

Then their (vector) sum $\mathbf u + \mathbf v$ is represented by the arrow $\vec {AC}$.


Also known as

A vector sum is also frequently seen referred to as a resultant.


Also see

  • Results about vector addition can be found here.


Sources