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:
- $\mathbf a + \mathbf b = 8 \mathbf i + \mathbf j$
Also see
Sources
- 1921: C.E. Weatherburn: Elementary Vector Analysis ... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Components of a Vector: $6$. Resolution of a vector
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $4$. Components of a Vector
- 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $1$ Vector Analysis $1.1$ Definitions, Elementary Approach: $(1.7 b)$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Graphical Representation of Complex Numbers. Vectors: $5 \ \text{(a)}$