How do you calculate the distance between 2 lines in 3 dimensional geometry? Watch this presentation to find out how to calculate the shortest distance between 2 parallel and non parallel lines. We also learn about skew lines and how to test if 2 lines are skew. This is a useful topic for students of grade 12 maths and students preparing for the NATA eligibility test in architecture. For more videos, visit my page https://www.mathmadeeasy.co/about https://www.mathmadeeasy.co/about-3

1. UCjmCXXIjd03JQad8-rUzs0Q
GRADE 12 MATHS
NATA eligibility test (mathematics)

2. Shortest distance
between 2 lines
Grade 12 maths Nata eligibility test for maths
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3. Here, we learn how to
• calculate the shortest distance between 2 lines
 What are skew lines?
 What is the shortest distance between 2 parallel lines?

5. Shortest distance between 2 lines 𝑟1 = 𝑎1 + 𝛼𝑏1
𝑟2 = 𝑎2+𝜇𝑏2
(𝑎2 − 𝑎1)(𝑏1 × 𝑏2)
𝑏1 × 𝑏2

6. Q 1 Find the shortest distance between
𝑟= ( 𝑖 + 2 𝑗 + 3 𝑘) + 𝛼 𝑖 − 3 𝑗 + 2 𝑘
and
𝑟 = (4 𝑖 + 5 𝑗+6 𝑘) + 𝜇(2 𝑖 + 3 𝑗 + 𝑘)

7. 𝑏1 × 𝑏2 =
𝑖 𝑗 𝑘
1 −3 2
2 3 1
= 𝑖 −3 − 6 − 𝑗 1 − 4 + 𝑘(3 + 6)
= −9 𝑖 + 3 𝑗 + 9 𝑘
𝑎2 − 𝑎1 = 3 𝑖 + 3 𝑗 + 3 𝑘

8. 𝑆ℎ𝑜𝑟𝑡𝑒𝑠𝑡 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 =
(3 𝑖 + 3 𝑗 + 3 𝑘)(−9 𝑖 + 3 𝑗 + 9 𝑘)
81 + 9 + 81
=
−27 + 9 + 27
3 19
=
3
19

9. Distance between 2 parallel lines 𝑟1 = 𝑎1 + 𝛼𝑏 and 𝑟2 = 𝑎2 + 𝜇𝑏
𝑑 =
𝑏 × (𝑎2 − 𝑎1)
𝑏
Skew lines are 2 lines that do not intersect
and are non parallel

11. Q 2
Find the shortest distance between the lines 𝑟 = 𝑖 + 2 𝑗 + 3 𝑘 + 𝛼( 𝑖 − 𝑗 + 𝑘)
and
𝑟=2 𝑖 − 𝑗 − 𝑘 + 𝜇(− 𝑖 + 𝑗 − 𝑘)

12. 𝑎2 − 𝑎1 = 𝑖 − 3 𝑗 − 4 𝑘
The 2 lines are parallel
𝑏 × 𝑎2 − 𝑎1 =
𝑖 𝑗 𝑘
1 −1 1
1 −3 −4
= 𝑖 4 + 3 − 𝑗 −4 − 1 + 𝑘(−3 + 1)
= 7 𝑖 + 5 𝑗 − 2 𝑘

14. Two non parallel lines
𝑥−𝑥1
𝑎1
=
𝑦−𝑦1
𝑏1
=
𝑧−𝑧1
𝑐1
𝑎𝑛𝑑
𝑥−𝑥2
𝑎2
=
𝑦−𝑦2
𝑏2
=
𝑧−𝑧2
𝑐2
𝑎𝑟𝑒 𝑠𝑘𝑒𝑤
𝑖𝑓𝑓
𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1
𝑎1 𝑏1 𝑐1
𝑎2 𝑏2 𝑐2
≠ 0

15. Q 3 Prove that the lines 2x = y = -z and x-1 = -y-2 = -2z +1 are skew

16. 𝑥
1
2
=
𝑦
1
=
𝑧
−1
𝑎𝑛𝑑
𝑥 − 1
1
=
𝑦 + 2
−1
=
𝑧 −
1
2
−1
2
Direction ratios of the 2 lines are <1,2,-2> and <2, -2, -1>

17. −1 2 −1
1 2 −2
2 −2 −1
= −1 −2 − 4 − 2 −1 + 4 − 1 −2 − 4 = 6 − 6 + 6 ≠ 0
The lines are skew

18. What did we learn?
How to calculate the shortest distance between 2 parallel and non parallel lines
How to check if 2 lines are skew

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