Question

Given points \( A(-3, 3), B(1, 1), C(1, -1), D(-2, -2) \), find the angle between diagonals \( AC \) and \( BD \).

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{2} \)
• C. \( \frac{\pi}{6} \)
• D. \( \frac{\pi}{3} \)

Hint:

Two vectors are perpendicular if their dot product is zero. Use coordinate subtraction to form vectors.

Answer 1

The Correct Option is B

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: \[ \vec{AC} \cdot \vec{BD} = 4(-3) + (-4)(-3) = -12 + 12 = 0 \Rightarrow \text{Vectors are perpendicular} \Rightarrow \theta = \frac{\pi}{2} \]