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For four points A, B, C and D in space, if the dot product of vectors AB and CD is given by k times [|AD|² + |BC|² - |AC|² - |BD|²], then what is the value of k?

1. 2
2. 1/3
3. 1/2
4. 1

Correct answer: 1/2

Solution

The dot product of vectors AB and CD can be expressed in terms of the lengths of the segments formed by the points A, B, C, and D. The relationship given simplifies to show that the coefficient k must be 1/2 to satisfy the equation, indicating that the geometric configuration of the points leads to this specific proportionality.

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