We are given the magnitudes of two vectors and their cross product: $$|\vec{a}| = 8, |\vec{b}| = 3, |\vec{a} \times \vec{b}| = 12$$ We know the definitions: $$|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta = 12$$ $$|\vec{a} \cdot \vec{b}| = |\vec{a}| |\vec{b}| \cos \theta$$
Substitute the given values into the cross product formula: $$8 \times 3 \times \sin \theta = 12$$ $$24 \sin \theta = 12$$ $$\sin \theta = \frac{12}{24} = \frac{1}{2}$$
Using the identity $\sin^2 \theta + \cos^2 \theta = 1$: $$\cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{1}{2}\right)^2 = 1 - \frac{1}{4} = \frac{3}{4}$$ $$\cos \theta = \frac{\sqrt{3}}{2}$$
Now substitute the values into the dot product formula: $$|\vec{a} \cdot \vec{b}| = |\vec{a}| |\vec{b}| \cos \theta$$ $$|\vec{a} \cdot \vec{b}| = 8 \times 3 \times \frac{\sqrt{3}}{2}$$ $$|\vec{a} \cdot \vec{b}| = 24 \times \frac{\sqrt{3}}{2} = 12\sqrt{3}$$
Final Answer: 12\sqrt{3}
AI generated content. Review strictly for academic accuracy.