​$$\text{If } |\vec{\alpha}| = 10, |\vec{\beta}| = 1 \text{ and } \vec{\alpha} \cdot \vec{\beta} = 6, \text{ then value of } |\vec{\alpha} \times \vec{\beta}| \text{ is}$$​​

Correct option is D

​$$\textbf{Given:} \\|\vec{\alpha}| = 10, \quad |\vec{\beta}| = 1, \quad \vec{\alpha} \cdot \vec{\beta} = 6 \\[6pt]\text{Use dot product formula:} \\\vec{\alpha} \cdot \vec{\beta} = |\vec{\alpha}| \cdot |\vec{\beta}| \cdot \cos\theta \\6 = 10 \cdot 1 \cdot \cos\theta \Rightarrow \cos\theta = \frac{6}{10} = 0.6 \\[6pt]\text{Find } \sin\theta: \\\sin^2\theta = 1 - \cos^2\theta = 1 - (0.6)^2 = 0.64 \Rightarrow \sin\theta = \sqrt{0.64} = 0.8 \\[6pt]\text{Use cross product magnitude formula:} \\|\vec{\alpha} \times \vec{\beta}| = |\vec{\alpha}| \cdot |\vec{\beta}| \cdot \sin\theta = 10 \cdot 1 \cdot 0.8 = 8 \\[10pt]\boxed{\text{Final Answer: } 8}$$​​