Equality of Vector Quantities
Theorem
Two vector quantities are equal if and only if they have the same magnitude and direction.
That is:
- $\mathbf a = \mathbf b \iff \paren {\size {\mathbf a} = \size {\mathbf b} \land \hat {\mathbf a} = \hat {\mathbf b} }$
where:
- $\hat {\mathbf a}$ denotes the unit vector in the direction of $\mathbf a$
- $\size {\mathbf a}$ denotes the magnitude of $\mathbf a$.
Proof
Let $\mathbf a$ and $\mathbf b$ be expressed in component form:
| \(\ds \mathbf a\) | \(=\) | \(\ds a_1 \mathbf e_1 + a_2 \mathbf e_2 + \cdots + a_n \mathbf e_n\) | ||||||||||||
| \(\ds \mathbf b\) | \(=\) | \(\ds b_1 \mathbf e_1 + b_2 \mathbf e_2 + \cdots + b_n \mathbf e_n\) |
where $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$ denote the unit vectors in the positive directions of the coordinate axes of the Cartesian coordinate space into which $\mathbf a$ has been embedded.
Thus $\mathbf a$ and $\mathbf b$ can be expressed as:
| \(\ds \mathbf a\) | \(=\) | \(\ds \tuple {a_1, a_2, \ldots, a_n}\) | ||||||||||||
| \(\ds \mathbf b\) | \(=\) | \(\ds \tuple {b_1, b_2, \ldots, b_n}\) |
By definition of vector length, we have that:
| \(\ds \size {\mathbf a}\) | \(=\) | \(\ds \size {\tuple {a_1, a_2, \ldots, a_n} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {\paren {a_1^2 + a_2^2 + \ldots + a_n^2} }\) |
and similarly:
| \(\ds \size {\mathbf b}\) | \(=\) | \(\ds \size {\tuple {b_1, b_2, \ldots, b_n} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {\paren {b_1^2 + b_2^2 + \ldots + b_n^2} }\) |
From Vector Quantity as Scalar Product of Unit Vector Quantity, it follows that:
| \(\ds \hat {\mathbf a}\) | \(=\) | \(\ds \widehat {\tuple {a_1, a_2, \ldots, a_n} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {\sqrt {\paren {a_1^2 + a_2^2 + \ldots + a_n^2} } } \mathbf a\) |
and similarly:
| \(\ds \hat {\mathbf b}\) | \(=\) | \(\ds \widehat {\tuple {b_1, b_2, \ldots, b_n} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {\sqrt {\paren {b_1^2 + b_2^2 + \ldots + b_n^2} } } \mathbf b\) |
Sufficient condition
Let $\mathbf a = \mathbf b$.
Then by Equality of Ordered Tuples:
- $(1): \quad a_1 = b_1, a_2 = b_2, \ldots a_n = b_n$
Then:
| \(\ds \size {\mathbf a}\) | \(=\) | \(\ds \sqrt {\paren {a_1^2 + a_2^2 + \ldots + a_n^2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {\paren {b_1^2 + b_2^2 + \ldots + b_n^2} }\) | from $(1)$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \size {\mathbf b}\) |
and:
| \(\ds \hat {\mathbf a}\) | \(=\) | \(\ds \dfrac 1 {\sqrt {\paren {a_1^2 + a_2^2 + \ldots + a_n^2} } } \mathbf a\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {\sqrt {\paren {b_1^2 + b_2^2 + \ldots + b_n^2} } } \mathbf b\) | from $(1)$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \hat {\mathbf b}\) |
Necessary Condition
Let $\size {\mathbf a} = \size {\mathbf b}$, and $\hat {\mathbf a} = \hat {\mathbf b}$.
Then:
| \(\ds \mathbf a\) | \(=\) | \(\ds \hat {\mathbf a} \sqrt {\paren {a_1^2 + a_2^2 + \ldots + a_n^2} }\) | from Vector Quantity as Scalar Product of Unit Vector Quantity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \hat {\mathbf a} \size {\mathbf a}\) | by definition of $\size {\mathbf a}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \hat {\mathbf b} \size {\mathbf b}\) | by assumption | |||||||||||
| \(\ds \) | \(=\) | \(\ds \hat {\mathbf b} \sqrt {\paren {b_1^2 + b_2^2 + \ldots + b_n^2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \mathbf b\) |
$\blacksquare$
Sources
- 1921: C.E. Weatherburn: Elementary Vector Analysis ... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Definitions: $3$. Definitions of terms
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $2$. Graphical Representation of Vectors
- 1961: I.M. Gel'fand: Lectures on Linear Algebra (2nd ed.) ... (previous) ... (next): $\S 1$: $n$-Dimensional vector spaces
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Fundamental Definitions: $1.$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): vector
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): vector
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 20$: Formulas from Vector Analysis: Fundamental Definitions: $1.$