Find the shortest distance between pair of lines: r → = i ^ + 2 j ^ + 3 k ^ + λ ( i ^ - 3 j ^ + 2 k ^ ) and r → = 4 i ^ + 5 j ^ + 6 k ^ + μ ( 2 i ^ + 3 j ^ + k ^ ) .

Comparing given lines with r 1 → = a 1 → + λ b 1 → and r 2 → = a 2 → + μ b 2 → Here, a 1 → = i + 2 j + 3 k ; b 1 → = i - 3 j + 2 k ∴ a 2 → = 4 i + 5 j + 6 k ; b 2 → = 2 i + 3 j + k Now b 1 → × b 2 → = i j k 1 - 3 2 2 3 1 = i ( - 3 - 6 ) - j ( 1 - 4 ) + k ( 3 + 6 ) = - 9 i + 3 j + 9 k = 3 ( - 3 i + j + 3 k ) ∴ b 1 → b 2 → = 3 9 + 1 + 9 = 3 19 Shortest distance between the lines = b → 1 × b → 2 . a → 2 - a → 1 b → 1 × b → 2 = 9 ( - 3 i + j + 3 k ) . ( i + j + k ) 3 19 = 3 19 | - 3 ( 1 ) + 1 ( 1 ) + 3 ( 1 ) | = 3 19 units

Find the shortest distance between pair of lines: r → = i ^ + 2 j ^ + 3 k ^ + λ ( i ^ - 3 j ^ + 2 k ^ ) and r → = 4 i ^ + 5 j ^ + 6 k ^ + μ ( 2 i ^ + 3 j ^ + k ^ ) .

Comparing given lines with r 1 → = a 1 → + λ b 1 → and r 2 → = a 2 → + μ b 2 → Here, a 1 → = i + 2 j + 3 k ; b 1 → = i - 3 j + 2 k ∴ a 2 → = 4 i + 5 j + 6 k ; b 2 → = 2 i + 3 j + k Now b 1 → × b 2 → = i j k 1 - 3 2 2 3 1 = i ( - 3 - 6 ) - j ( 1 - 4 ) + k ( 3 + 6 ) = - 9 i + 3 j + 9 k = 3 ( - 3 i + j + 3 k ) ∴ b 1 → b 2 → = 3 9 + 1 + 9 = 3 19 Shortest distance between the lines = b → 1 × b → 2 . a → 2 - a → 1 b → 1 × b → 2 = 9 ( - 3 i + j + 3 k ) . ( i + j + k ) 3 19 = 3 19 | - 3 ( 1 ) + 1 ( 1 ) + 3 ( 1 ) | = 3 19 units

Find the shortest distance between the lines x - 1 2 = y - 2 3 = z - 3 4 and x - 2 3 = y - 4 4 = z - 5 5 Also find the equations of this line.

The given lines are x - 1 2 = y - 2 3 = z - 3 4 . . . . . . . . . . . . . . . . . . . . . . . ( 1 ) x - 2 3 = y - 4 4 = z - 5 5 . . . . . . . . . . . . . . . . . . . . . . . . . ( 2 ) Any point on line ( 1 ) be M ( 2 t + 1 , 3 t + 2 , 4 t + 3 ) and any point on line ( 2 ) be N ( 3 s + 2 , 4 s + 4 , 5 s + 5 ) ∴ Direction ratios of line M N are < 3 s + 2 - 2 t - 1 , 4 s + 4 - 3 t - 2 , 5 s + 5 - 4 t - 3 > = < 3 s + 2 - 2 t - 1 , 4 s + 4 - 3 t + 2 , 5 s - 4 t - 3 s > = < 3 s - 2 t + 1 , 4 s - 3 t + 2 , 5 s - 4 t + 2 > M N will be the shortest distance between ( 1 ) and ( 2 ) iff M N be perpendicular to line ( 1 ) and ( 2 ) . ∴ 2 ( 3 s - 2 t + 1 ) + 3 ( 4 s - 3 t + 2 ) + 4 ( 5 s - 4 t + 2 ) = 0 and 3 ( 3 s - 2 t + 1 ) + 4 ( 4 s - 3 t + 2 ) + 5 ( 5 s - 4 t + 2 ) = 0 38 s - 29 t + 16 = 0 . . . . . . . . . . . . . . . ( 3 ) 50 s - 38 r + 21 = 0 . . . . . . . . . . . . . . . . ( 4 ) On solving ( 3 ) and ( 4 ) , we have s - 309 + 608 = t 800 - 798 = 1 - 1444 + 1450 ∴ s - 1 = t 2 = 1 6 ⇒ s = - 1 6 ; t = 1 3 Any point on lines ( 1 ) and ( 2 ) are M 5 3 , 3 , 13 3 and N 3 2 , 10 3 , 25 6 ∴ Shortest distance between given lines M N = 3 2 - 5 3 2 + 10 3 - 3 2 + 25 6 - 13 3 2 = 1 36 + 1 9 + 1 36 = 6 36 = 1 6 units The equations of line M N are: x - 5 3 3 2 - 5 3 = y - 3 10 3 - 3 = z - 13 3 25 6 - 13 3 i . e x - 5 3 - 1 6 = y - 3 1 3 = z - 13 3 - 1 6 i . e . x - 5 3 1 = y - 3 - 2 = z - 13 3 1

Determine (by computing S.D.) whether the pair of lines intersect: r → = i ^ - j ^ + λ ( 2 i ^ + k ^ ) ; r ^ = 2 i ^ - j ^ + μ ( i ^ + j ^ - k ^ )

Given lines are r → = ( i - j ) + λ ( 2 i + k ) . . . . . . . . . . . ( 1 ) r → = ( 2 i - j ) + μ ( i + j - k ) . . . . . . . . . . . . . . . ( 2 ) If shortest distance is not zero, it means lines do not intersect. Comparing equations ( 1 ) and ( 2 ) with r → = a → 1 + λ b → 1 ; r = a → 2 + λ b → 2 where, a → 1 = i - j ; b → 1 = 2 i + k ; a → 2 = 2 i - j ; b → 2 = i + j - k Here a → 2 - a → 1 = 2 i - j - i + j = i + 0 j and b → 1 × b → 2 = i j k 2 0 1 1 1 - 1 = i ( 0 - 1 ) - j ( - 2 - 1 ) + k ( 2 - 1 i + 3 j + 2 k ) Here, ( a → 2 - a → 1 ) · ( b → 1 × b → 2 ) = ( i + 0 k ) · ( - i + 3 j + 2 k ) = - 1 ≠ 0 Thus given lines do not intersects.

Determine (by computing S.D.) whether pair of lines intersect: r → = + j ^ - k + λ ( 3 i ^ - j ^ ) ; r ^ = 4 i ^ - k ^ + μ ( 2 i ^ + 3 k ^ )

Given lines are, r → = ( i + j - k ) + λ ( 3 i - j ) . . . . . . . . . . . . ( 1 ) r → = ( 4 i - k ) + ( 2 i + 3 k ) . . . . . . . . . . . . . ( 2 ) If shortest distance is zero, it means lines intersect. Comparing equation ( 1 ) and equation ( 2 ) with r → = a → 1 + λ b → 2 a → 1 = i + j - k ; b → 1 = 3 i - j ; a → 2 = 4 i + 0 j - k ; b → 2 = 2 i + 0 j + 3 k a → 2 - a → 1 = ( 4 i + 0 j - k ) - ( i + j - k ) = 3 i - j + 0 k b → 1 × b → 2 = i j k 3 - 1 0 2 0 3 = i ( - 3 - 0 - j ( 9 - 0 ) + k ( 0 + 2 ) = - 3 i + 9 j + 2 k ∴ ( a → 2 - a → 1 ) · ( b → 1 × b → 2 ) = ( 3 i - j + 0 k ) . ( - 3 i - 9 j + 2 k ) = 3 ( - 3 ) - 1 ( - 9 ) + 0 ( 2 ) = - 9 + 9 = 0 Shortest distance = ( a → 2 - a → 1 ) . ( b → 1 × b → 2 ) b → 1 × b → 2 = 0 Hence, the given lines are intersecting.

Determine (by computing shortest distance) whether or not the lines x - 1 2 = y + 1 3 = z ; x + 1 5 = y - 2 1 , z = 2 intersect.

Given lines in vector form. r → = i - j + 0 k + λ ( 2 i + 3 j + k ) r → = - i + 2 j + 2 k + μ ( 5 i + j + 0 k ) On comparing with r → = a → 1 + λ b → 1 and r → 2 = a → 2 + μ b → 2 ∴ a → 1 = i - j + 0 k ; b → 1 = 2 i + 3 j + k ; a 2 → = - i + 2 j + 2 k ; b 2 → = 5 i + j + 0 k ∴ a 2 → - a 1 → = - 2 i + 3 j + 2 k b 1 → × b 2 → = i j k 2 3 1 5 1 0 = i ( 0 - 1 ) - j ( 0 - 5 ) + k ( 2 - 15 ) = - i + 5 j - 13 k Here ( a 2 → - a 1 → ) · ( b 1 → × b 2 → ) = ( - 2 i + 3 j + 2 k ) . ( - i + 5 j - 13 k ) = - 2 ( - 1 ) + 3 ( 5 ) + 2 ( - 13 ) = 2 + 15 - 26 = - 9 ≠ 0 Shortest distance is non-zero. Hence, given lines do not intersect.