Direction Ratios And Angle Between Lines
Two lines have direction ratios (1, 2, 2) and (2, -2, 1) respectively, so what is the acute angle between these two lines?
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About This Question
- Subject
- mathematics
- Chapter
- three dimensional geometry
- Topic
- direction ratios and angle between lines
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
cos−1(0)
The method here is the dot-product formula for the angle between two lines given their direction ratios: \cos\theta = |a_1 a_2 + b_1 b_2 + c_1 c_2| / (\sqrt{a_1^2+b_1^2+c_1^2},\sqrt{a_2^2+b_2^2+c_2^2}). This is the canonical JEE Advanced tool for measuring inclination between lines in space. Computing the numerator: (1)(2) + (2)(-2) + (2)(1) = 2 - 4 + 2 = 0. The denominators are \sqrt{1+4+4} = 3 and \sqrt{4+4+1} = 3, giving \cos\theta = 0/9 = 0. Therefore the lines are perpendicular and \theta = \cos^{-1}(0) = 90 degrees. Option \cos^{-1}(4/9) ignores the cancellation in the numerator. Option \cos^{-1}(2/9) takes only the positive contributions. Option \cos^{-1}(1/3) mismatches the magnitudes. This relies on the standard perpendicularity criterion a_1 a_2 + b_1 b_2 + c_1 c_2 = 0. Plausibility check: a zero dot product of the direction vectors geometrically guarantees orthogonality, consistent with the obtained right angle.
This easy difficulty mathematics question is from the chapter three dimensional geometry, covering the topic of direction ratios and angle between lines. It appeared in the 2025 exam.
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