Geometry Level 3

Let $$\Delta PQR$$ be a triangle. Let $$\vec{a}=\vec{QR}, \vec{b}=\vec{RP}$$ and $$\vec{c}=\vec{PQ}$$. If $$|\vec{a}|=12,.|\vec{b}|=4\sqrt{3}$$ and $$\vec{b} \cdot \vec{c}=24$$, then which of the following is(are) true $\begin{array}{} (1) \, \frac{|\vec{c}|^2}{2}-|\vec{a}|=12 \quad \quad \quad \quad & (2) \, \frac{|\vec{c}|^2}{2}+|\vec{a}|=12 \\ (3) \, |\vec{a} \times \vec{b}+\vec{c} \times \vec{a}|=48\sqrt{3}& (4) \, \vec{a} \cdot \vec{b}=-72 \end{array}$
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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