3 D GEOMETRY - ANGLE BETWEEN 2 LINES
- 1. UCjmCXXIjd03JQad8-rUzs0Q GRADE 12 MATHS NATA eligibility test (mathematics)
- 2. Angle between 2 lines 3 dimensional geometry
- 3. How do you calculate the angle between 2 lines? Why do we need it? This is a part of 3 dimensional geometry and used in grade 12 mathematics Also used in the maths section of the NATA eligibility test in architecturee
- 4. We learn to calculate Angle between 2 lines Equation of a line passing through a point and perpendicular to 2 lines How to calculate the foot of a perpendicular from a point to a line
- 6. Given 2 lines in Cartesian form 𝑥 − 𝑥1 𝑎1 = 𝑦 − 𝑦1 𝑏1 = 𝑧 − 𝑧1 𝑐1 and 𝑥 − 𝑥2 𝑎2 = 𝑦 − 𝑦2 𝑏2 = 𝑧 − 𝑧2 𝑐2 Angle between the 2 lines 𝑐𝑜𝑠𝜃 = 𝑎1 𝑎2+𝑏1 𝑏2+𝑐1 𝑐2 𝑎1 2+𝑏1 2+𝑐1 2 𝑎2 2+𝑏2 2+𝑐2 2 Acute angle 𝑐𝑜𝑠𝜃 = 𝑎1 𝑎2 + 𝑏1 𝑏2 + 𝑐1 𝑐2 𝑎1 2 + 𝑏1 2 + 𝑐1 1 𝑎2 2 + 𝑏2 2 + 𝑐2 2
- 7. 𝐼𝑓 < 𝑙1, 𝑚1, 𝑛1 >, < 𝑙2, 𝑚2, 𝑛2 > 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑐𝑜𝑠𝑖𝑛𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑐𝑜𝑠𝜃 = 𝑙1 𝑙2 + 𝑚1 𝑚2 + 𝑛1 𝑛2
- 8. Q 1 Find the angle between the lines 𝑟=(3 𝑖 − 2 𝑗 + 6 𝑘) + 𝛿(2 𝑖 + 𝑗 + 2 𝑘) 𝑎𝑛𝑑 𝑟 = 2 𝑗 − 5 𝑘 + 𝜎(6 𝑖 + 3 𝑗 + 2 𝑘)
- 9. Converting to Cartesian form 𝑥 − 3 2 = 𝑦 + 2 1 = 𝑧 − 6 2 𝑥 6 = 𝑦 − 2 3 = 𝑧 + 5 2 𝑐𝑜𝑠𝜃 = 12 + 3 + 4 4 + 1 + 4 36 + 9 + 4 = 19 21 𝜃=𝑐𝑜𝑠−1 ( 19 21 )
- 10. Equation of a line passing through a point 𝑎 𝑎𝑛𝑑 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑡𝑜 the lines 𝑟1 = 𝑎1 + 𝛼𝑏1 𝑎𝑛𝑑 𝑟2 = 𝑎2 + 𝜇𝑏2 𝑟=𝑎 +𝜌(𝑏1 × 𝑏2)
- 12. Q 2) Find the equation of the line passing through (1,2,-4) and perpendicular to the lines 𝑟=(8 𝑖 − 19 𝑗 + 10 𝑘) + 𝛼(3 𝑖 − 16 𝑗 + 7 𝑘) 𝑎𝑛𝑑 𝑟 = 15 𝑖 + 29 𝑗 + 5 𝑘 + 𝜇(3 𝑖 + 8 𝑗 − 5 𝑘)
- 13. The required line is perpendicular to the 2 lines 𝑖 𝑗 𝑘 3 −16 7 3 8 −5 = 𝑖 80 − 56 − 𝑗 −15 − 21 + 𝑘(24 + 48) = 24 𝑖 + 36 𝑗 + 72 𝑘
- 14. Equation of the required line 𝑖+2 𝑗 − 4 𝑘 + 𝑡(24 𝑖 + 36 𝑗 + 72 𝑘) = 1 + 24𝑡 𝑖 + 2 + 36𝑡 𝑗 + (−4 + 72𝑡) 𝑘
- 15. 𝑥 + 3 5 = 𝑦 − 1 2 = 𝑧 + 4 3 Q 3 Find the foot of the perpendicular drawn from (0,2,3) on the line
- 16. 𝑙𝑒𝑡 𝑥 + 3 5 = 𝑦 − 1 2 = 𝑧 + 4 3 = 𝑡 x = 5t -3 , y = 2t +1, z = 3t -4 (0,2,3) (5t -3, 2t+1,3t-4) Direction ratios of the perpendicular < 5t-3, 2t-1, 3t-7> Direction ratios of the line =<5,2,3>
- 17. Since the 2 lines are perpendicular 5(5t-3)+2(2t-1)+3(3t-7)=0 t = 1 Point of intersection (2,3, -1)
- 18. What have we learnt? How to calculate angle between 2 lines How to calculate equation of a line passing through a point and perpendicular to 2 lines How to calculate foot of the perpendicular from a point to a line
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