Question

Find the value of p if the shortest distance between the lines

`vec r = (hat i + 2 hat j + hat k) + λ (hat i - hat j + hat k) and vec r = (p hat i - hat j - hat k) + µ (2 hat i + hat j + 2 hat k) "is" 3/sqrt2` units.

Answer 1

The equations of the lines are given in the form `vec r = vec a + λ vec b` where `vec a` is a point on the line and `vec b` is the direction vector.

Line 1: `vec r = (hat i + 2 hat j + hat k) + λ (hat i - hat j + hat k)`

`a_1 = hat i + 2 hat j + hat k`

`b_1 = hat i - hat j + hat k`

Line 2: `vec r = (p hat i - hat j - hat k) + µ (2 hat i + hat j + 2 hat k)`

`a_2 = p hat i - hat j - hat k`

`b_2 = 2 hat i + hat j + 2 hat k`

The shortest distance d is given by:

d = `|((vec a_2 - vec a_1) . (vec b_1 xx vec b_2))/|vec b_1 xx b_2||`

Find `(vec a_2 - vec a_1)`

`(vec a_2 - vec a_1) = (p - 1) hat i + (-1 - 2) hat j + (-1 - 1) hat k`

= `(p - 1)hat i - 3 hat j - 2 hat k`

Find `(vec b_1 xx vec b_2)`

`vec b_1 xx vec b_2 = |(hat i, hat j, hat k),(1, -1, 1),(2, 1, 2)|`

= `hat i(-2 - 1) - hat j(2 - 2) + hat k(1 - (-2))`

= `3 hat i + 0 hat j + 3 hat k`

Find the magnitude `|vec b_1 xx vec b_2|`

`|vec b_1 xx vec b_2| = sqrt((-3)^2 + 0^2 + 3^2)`

= `sqrt(9 + 9)`

= `sqrt18`

= `3 sqrt2`

Solve for p:

Substitute the components and the given distance d = `3/sqrt2` into the formula:

`3/sqrt2 = |(((p - 1)hat i - 3 hat j - 2 hat k) . (-3 hat i + 3 hat k))/(3 sqrt2)|`

`3/sqrt2 = |-3(p - 1) + 0 - 6|/(3 sqrt2)`

9 = |−3p + 3 − 6|   ...[Multiply both sides by `3 sqrt2`]

9 = |−3p − 3|

Taking the absolute value into account, we have two cases:

−3p − 3 = 9

−3p = 9 + 3

−3p = 12

p = `12/-3`

p = −4

−3p − 3 = −9

−3p = −9 + 3

−3p = −6

p = `(-6)/(-3)`

p = 2

The values of p are 2 or −4