The angle between the line joining the points A(1, 2, 3) and B(3, 4, 4) and the line joining the points C(1, 0, −2) and D(3, −1, 0) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\cos^{-1}\left(\displaystyle\frac{4}{9}\right)\)
The direction ratios of each line are the differences of the coordinates of its two points:
AB: \(3-1,\ 4-2,\ 4-3 = 2, 2, 1\)
CD: \(3-1,\ -1-0,\ 0-(-2) = 2, -1, 2\)
Both have length \(\sqrt{9} = 3\).
\( \displaystyle \cos\theta = \frac{|(2)(2)+(2)(-1)+(1)(2)|}{3 \times 3} = \frac{4}{9} \)
So the angle is \(\cos^{-1}\left(\frac{4}{9}\right)\).
Why the other options are incorrect
Option (1) \(\cos^{-1}\left(\displaystyle\frac{1}{9}\right)\) — Incorrect. The dot product of the directions is 4, not 1.
Option (2) \(\cos^{-1}\left(\displaystyle\frac{8}{9}\right)\) — Incorrect. This drops the minus sign in \((2)(-1)\): \(4+2+2 = 8\).
Option (4) \(\displaystyle\frac{\pi}{2}\) — Incorrect. The dot product 4 is not zero, so the lines are not perpendicular.