Let \(PQR\) be a triangle. Let \(\vec{a} = \overline{QR}, \vec{b} = \overline{RP}, \text{ and } \vec{c} = \overline{PQ}\). If \(|\vec{a}| = 12\), \(|\vec{b}| = 4\sqrt{3}\), and \(\vec{b} \cdot \vec{c} = 24\), then which of the following is (are) true?

Question

\(\frac{1}{2}|\vec{c}|^2 - |\vec{a}| = 12\)

\(\frac{1}{2}|\vec{c}|^2 + |\vec{a}| = 30\)

\(|\vec{a} \times \vec{b} + \vec{c} \times \vec{a}| = 48\sqrt{3}\)

\(\vec{a} \cdot \vec{b} = -72\)

Solution

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