Finding Shortest Distance Between Skew Lines and Co-ordinates
Explore the concept of skew lines and calculate the shortest distance between two non-intersecting lines. Understand the methodology to determine the coordinates and distance efficiently for problem-solving.
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Skew Lines The two given lines given are skew (i.e. they do not meet). What is the shortest distance between them? ? ? ? ??= + ? ? ? ? ? ?? ? ?? ? ? ? ??= + ? shortest distance ? Also find the co-ordinates of ? and ?.
?? must be perpendicular to both lines. ? ? ? Let ?? = ? ? ? 2 Then = 0 2? 3? + 2? = 0 (1) . 3 2 ? ? ? 2 2 1 And = 0 2? + 2? ? = 0 (2) . We have two equations but three unknowns so they cannot be solved but we don t need to, we only need find the relative sizes of them. Adding equations (1) and (2) gives ? = ? and substitution gives ? = 2?. 1 2 2 So with magnitude 3? where ? is some number. ?? = ?
Let?and ? be general points on ?1 and ?2, respectively. The coordinates of each are: ? = 5 + 2?, 4 3?,5 + 2? ? = 10 2?,2 + 2?,10 ? . The vector joining these points is: (10 2?) 5 + 2? (2 + 2?) 4 3? (10 ?) 5 + 2? 5 2? 2? 6 + 2? + 3? 5 ? 2? ? ? = = This must be parallel to ?? so: 5 2? 2? 6 + 2? + 3? 5 ? 2? 1 2 2 = ? This gives us three equations with three unknowns which we can solve.
(1) (2) (3) 5 6 5 2? +2? ? 2? +3? 2? = = = ? 2? 2? 1 + (2) (4) 2(3) (1) (5) 4 (5) (6) = = = 3? 3? 0 11 5 6 +? 2? +3? Yielding: ? = 2 ? = 3 ? = 3 Substituting these values gives us the required solution to our problem: ? = 5 + 2( 2), 4 3( 2),5 + 2( 2)= ?,?,? Q = 10 2(3),2 + 2(3),10 (3) = ?,?,? The shortest distance = 3? = ?
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ? ? ??= + ? ? ? ? ? ?? ? ? ??= + ? shortest distance ? ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ? ? ??= + ? ? ? ? ?? ? ? ??= + ? ? ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ? ? ??= + ? ? ? ?? ? ? ? ? ? ??= + ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ? ? ??= + ? ? ? ? ? ? ?? ?? ? ??= + ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ? ? ??= + ? ? ? ? ?? ? ? ??= + ? ? ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ? ? ??= + ? ? ? ? ?? ? ? ??= + ? ? ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ? ? ??= + ? ? ? ? ?? ? ? ??= + ? ? ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ? ? ??= + ? ? ? ?? ? ? ? ? ? ??= + ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ? ? ??= + ? ? ? ? ? ? ?? ?? ? ??= + ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ? ? ??= + ? ? ? ? ?? ? ? ??= + ? ? ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ??= + ? ? ? ? ? ? ?? ? ? ??= + ? ? ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ??= + ? ? ? ? ? ? ? ?? ? ? ??= + ? shortest distance ? ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ??= + ? ? ? ? ? ?? ? ? ? ? ? ??= + ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ??= + ? ? ? ? ? ? ? ? ?? ?? ? ??= + ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ??= + ? ? ? ? ? ? ?? ? ? ??= + ? ? ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ? ? ? ? ??= + ? ? ?? ? ? ??= + ? ? ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ? ? ? ? ??= + ? ? ? ?? ? ? ??= + ? shortest distance ? ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ? ? ? ? ??= + ? ?? ? ? ? ? ? ??= + ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ? ? ? ? ??= + ? ? ?? ? ? ??= + ? ? ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
Skew Lines The two given lines given are skew (i.e they do not meet). What is the shortest distance between them? ? ? ? ? ? ? ? ??= + ? ? ?? ? ? ??= + ? ? ? shortest distance ? Also find the co-ordinates of ? and ?. SIC_27
