Definition:Vector Sum/Component Definition

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

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

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

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.


Example $1$


\(\ds \mathbf a\) \(=\) \(\ds 6 \mathbf i + 4 \mathbf j + 3 \mathbf k\)
\(\ds \mathbf b\) \(=\) \(\ds 2 \mathbf i - 3 \mathbf j - 3 \mathbf k\)


$\mathbf a + \mathbf b = 8 \mathbf i + \mathbf j$

Also see