Question

If the coordinates of the points \( A \), \( B \), \( C \), and \( D \) are \( (1, 2, 3) \), \( (4, 5, 7) \), \( (-4, 3, -6) \), and \( (2, 9, 2) \) respectively, then find the angle between the lines AB and CD.

Hint:

To find the angle between two vectors, use the dot product formula and calculate the cosine of the angle between them.

Answer 1

Step 1: Find the direction vectors of lines AB and CD: \[ \vec{AB} = B - A = (4 - 1, 5 - 2, 7 - 3) = (3, 3, 4), \] \[ \vec{CD} = D - C = (2 - (-4), 9 - 3, 2 - (-6)) = (6, 6, 8). \] 

Step 2: Use the dot product formula to find the cosine of the angle \( \theta \) between the two vectors: \[ \cos \theta = \frac{\vec{AB} \cdot \vec{CD}}{|\vec{AB}| |\vec{CD}|}. \] First, compute the dot product: \[ \vec{AB} \cdot \vec{CD} = (3 \times 6) + (3 \times 6) + (4 \times 8) = 18 + 18 + 32 = 68. \] Now, compute the magnitudes of \( \vec{AB} \) and \( \vec{CD} \): \[ |\vec{AB}| = \sqrt{3^2 + 3^2 + 4^2} = \sqrt{9 + 9 + 16} = \sqrt{34}, \] \[ |\vec{CD}| = \sqrt{6^2 + 6^2 + 8^2} = \sqrt{36 + 36 + 64} = \sqrt{136}. \] 

Step 3: Substitute the values into the formula for \( \cos \theta \): \[ \cos \theta = \frac{68}{\sqrt{34} \times \sqrt{136}} = \frac{68}{\sqrt{4624}} = \frac{68}{68} = 1. \] Thus, the angle between the lines is \( \theta = 0^\circ \).