What does the term 'unique' refers to in vectors??

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​$$\begin{gathered} \mathrm{Unique} \mathrm{means} \mathrm{one} \mathrm{of} \mathrm{its} \mathrm{kind}. \\ \mathrm{e}.\mathrm{g}. \mathrm{Cross} \mathrm{product} \mathrm{of} \mathrm{two} \mathrm{vectors} \overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}} \mathrm{is} \mathrm{unique} \mathrm{which} \mathrm{is} \mathrm{perpendicular} \\ \mathrm{to} \mathrm{both} \mathrm{the} \mathrm{vectors}. \\ \mathrm{Also} \mathrm{line} \mathrm{passing} \mathrm{through} \mathrm{two} \mathrm{points} \mathrm{say} \left(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\right) \mathrm{and} \left(2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+2\hat{\mathrm{k}}\right) \mathrm{will} \mathrm{be} \\ \mathrm{unique} \mathrm{i}.\mathrm{e}. \mathrm{there} \mathrm{will} \mathrm{exist} \mathrm{one} \mathrm{one} \mathrm{line} \mathrm{that} \mathrm{contains} \mathrm{both} \mathrm{the} \mathrm{points}. \mathrm{Equation} \mathrm{of} \mathrm{this} \\ \mathrm{unique} \mathrm{line} \mathrm{will} \mathrm{be} \\ \overrightarrow{\mathrm{r}}=\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\right) +\mathrm{t}\left[\left(2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+2\hat{\mathrm{k}}\right)-\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)\right], \mathrm{where} \mathrm{t} \mathrm{is} \mathrm{the} \mathrm{parameter}. \\ \mathbf{\Rightarrow }\overrightarrow{\mathbf{r}}\mathbf{=}\left(\hat{\mathbf{i}}\mathbf{+}\hat{\mathbf{j}}\mathbf{+}\hat{\mathbf{k}}\right)\mathbf{+}\mathbf{t}\left(\hat{\mathbf{i}}\mathbf{+}\mathbf{2}\hat{\mathbf{j}}\mathbf{+}\hat{\mathbf{k}}\right) \end{gathered}$$

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