Question

Find the angle between the following pairs of lines: (i) (ii)

Answer 1

(i)

The pair of lines is given as,

x−2 2 = y−1 5 = z+3 −3

And,

x+2 −1 = y−4 8 = z−5 4

Consider that Q is the angle between the given lines.

The formula to calculate the angle between the pair of lines is,

cosQ=| b 1 → ⋅ b 2 → | b 1 → || b 2 → | |(1)

The given lines are parallel to the vectors,

b 1 → =2 i ^ +5 j ^ −3 k ^

And,

b 2 → =− i ^ +8 j ^ +4 k ^

The value of b 1 → is,

b 1 → = 2 2 + 5 2 + ( −3 ) 2 = 38

The value of b 2 → is,

b 2 → = ( −1 ) 2 + 8 2 + 4 2 = 81 =9

The value of b 1 → ⋅ b 2 → is,

b 1 → ⋅ b 2 → =( 2 i ^ +5 j ^ −3 k ^ )⋅( − i ^ +8 j ^ +4 k ^ ) =2×( −1 )+5×8+( −3 )×4 =−2+40−12 =26

Substitute the values in equation (1),

cosQ=| 26 9 38 | Q= cos −1 ( 26 9 38 )

(ii)

The pair of lines is given as,

x 2 = y 2 = z 1

And,

x−5 4 = y−2 1 = z−3 8

Consider that Q is the angle between the given lines.

The formula to calculate the angle between the pair of lines is,

cosQ=| b 1 → ⋅ b 2 → | b 1 → || b 2 → | |(1)

The given lines are parallel to the vectors,

b 1 → =2 i ^ +2 j ^ + k ^

And,

b 2 → =4 i ^ + j ^ +8 k ^

The value of b 1 → is,

b 1 → = 2 2 + ( 2 ) 2 + ( 1 ) 2 = 9 =3

The value of b 2 → is,

b 2 → = 4 2 + ( 1 ) 2 + ( 8 ) 2 = 81 =9

The value of b 1 → ⋅ b 2 → is,

b 1 → ⋅ b 2 → =( 2 i ^ +2 j ^ + k ^ )⋅( 4 i ^ + j ^ +8 k ^ ) =2×4+2×( 1 )+1×( 8 ) =8+2+8 =18

Substitute the values in equation (1),

cosQ=| 18 3×9 | cosQ= 2 3 Q= cos −1 ( 2 3 )