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If \(\vec{A}\cdot\vec{B}=3AB\), then the value of \(|\vec{A}+\vec{B}|\) is:

1. \(\sqrt{A^2+B^2+3AB}\)
2. \(A+B\)
3. \(\sqrt{A^2+B^2-3AB}\)
4. \(\sqrt{A^2+B^2+AB}\)

Correct answer: \(\sqrt{A^2+B^2+3AB}\)

Solution

We use \(|\vec{A}+\vec{B}|^2=|\vec{A}|^2+|\vec{B}|^2+2\vec{A}\cdot\vec{B}\). Given \(\vec{A}\cdot\vec{B}=3AB\), the expression becomes \(A^2+B^2+6AB\), so the OCR text is likely corrupted. The intended option from the source appears to be the square-root form matching the sum magnitude formula.

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