Question:
Given points \( A(-3, 3), B(1, 1), C(1, -1), D(-2, -2) \), find the angle between diagonals \( AC \) and \( BD \).
Given points \( A(-3, 3), B(1, 1), C(1, -1), D(-2, -2) \), find the angle between diagonals \( AC \) and \( BD \).
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Two vectors are perpendicular if their dot product is zero. Use coordinate subtraction to form vectors.
Updated On: May 17, 2025
- \( \frac{\pi}{4} \)
- \( \frac{\pi}{2} \)
- \( \frac{\pi}{6} \)
- \( \frac{\pi}{3} \)
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The Correct Option is B
Solution and Explanation
Let vectors:
- \( \vec{AC} = C - A = (1 + 3, -1 - 3) = (4, -4) \)
- \( \vec{BD} = D - B = (-2 - 1, -2 - 1) = (-3, -3) \)
Find dot product:
\[
\vec{AC} \cdot \vec{BD} = 4(-3) + (-4)(-3) = -12 + 12 = 0
\Rightarrow \text{Vectors are perpendicular}
\Rightarrow \theta = \frac{\pi}{2}
\]
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