Geometry Level 3

In $$\mathbb R^2$$ if the magnitude of the projection vector of the vector $$\alpha \hat +\beta \hat j$$ on $$\sqrt3 \hat i +\hat j$$ is $$\sqrt3$$ and if $$\alpha=2+\sqrt3 \beta$$, then find the possible value(s) of $$|\alpha|$$. $\begin{array}{} (1) \, 1 \quad \quad \quad \quad \quad \quad \quad \quad & (2) \, 2 \\ (3) \, 3 & (4) \, 4 \end{array}$

Note:

• Submit your answer as the increasing order of the serial numbers of all the correct options.

• For eg, if your answer is $$(1),(2)$$, then submit 12 as the correct answer, if your answer is $$(2),(3),(4)$$, then submit 234 as the correct answer.

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