Equality of Vector Quantities

Theorem
Two vector quantities are equal 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:

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:

By definition of vector length, we have that:

and similarly:

From Vector Quantity as Scalar Product of Unit Vector Quantity, it follows that:

and similarly:

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:

and:

Necessary Condition
Let $\size {\mathbf a} = \size {\mathbf b}$, and $\hat {\mathbf a} = \hat {\mathbf b}$.

Then: