Write the inequality $\mathbf a < \mathbf b$ with unit vectors?

Question

Say I have the vectors $\mathbf{a}\in \mathbb R^3$ and $\mathbf b\in \mathbb R^3$, and I write \begin{align} \mathbf a &< \mathbf b\tag 1\\ a_j &< b_j \qquad j = 1,2,3 \tag 2 \end{align}

My question:

Is it correct to write the inequality $(1)$ with unit vectors? I mean the following, if \begin{align} \mathbf a &= \hat e_1 a_1 + \hat e_2 a_2 \tag 3 +\hat e_3 a_3\\ \mathbf b &= \hat e_1 b_1 + \hat e_2 b_2 +\hat e_3 b_3\tag 4 \end{align} Is $\mathbf a < \mathbf b$ equivalent to the following \begin{align} \hat e_1 a_1 + \hat e_2 a_2 +\hat e_3 a_3&< \hat e_1 b_1 + \hat e_2 b_2 +\hat e_3 b_3\tag 5\\ &\iff \\ \hat e_1 (a_1 < b_1) + \hat e_2 (a_2 &< b_2) + \hat e_3 (a_3 < b_3) \tag 6\\ &\iff \\ \hat e_1 (a_1 - b_1 < 0) + \hat e_2 (a_2 - b_2 &< 0) + \hat e_3 (a_3-b_3<0)\tag 7 \end{align} ?

Answer 1

Your (5), (6), (7) is not mathematical notation. Instead, write

\begin{align} \hat e_1 a_1 + \hat e_2 a_2 +\hat e_3 a_3&< \hat e_1 b_1 + \hat e_2 b_2 +\hat e_3 b_3\tag 5\\ &\iff \\ a_1 < b_1 \quad\text{and}\quad a_2 &< b_2 \quad\text{and}\quad a_3 < b_3 \tag 6\\ &\iff \\ a_1 - b_1 < 0 \quad\text{and}\quad a_2 - b_2 &< 0 \quad\text{and}\quad a_3-b_3<0\tag 7 \end{align}

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Also note: You should state that (2) is the definition of (1), since it is not something that is commonly used.