# Definition:Vector Sum/Component Definition

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## Definition

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.

## Examples

### Example $1$

Let:

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

Then:

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