(II \( \widehat{\mathrm{A}}=(-1,2,2), \widehat{\mathrm{B}} \) is a unit vector then \( \vec{A} \cdot \vec{B} \) could be equal to \( \begin{array}{llll}\text { (a) } 7 & \text { (b) } 4 & \text { (c) } 2 & \text { (d) }-4\end{array} \)
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To find the possible values of \( \vec{A} \cdot \vec{B} \), where \( \vec{A} = (-1,2,2) \) and \( \widehat{\mathrm{B}} \) is a unit vector, we start by noting that the dot product \( \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta \). Here, \( |\vec{B}| = 1 \). First, we compute the magnitude of \( \vec{A} \): \[ |\vec{A}| = \sqrt{(-1)^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \] So, \[ \vec{A} \cdot \vec{B} = 3 \cos \theta \] Since \( \cos \theta \) can range from -1 to 1, the dot product can vary from: \[ -3 \leq \vec{A} \cdot \vec{B} \leq 3 \] Now reviewing the answer choices: - \( 7 \): Out of range - \( 4 \): Out of range - \( 2 \): In range - \( -4 \): Out of range Therefore, the only possible value from the supplied options is \( 2 \).
