Question

(a) Find the velocity of Q relative to P. (b) Find the value of λ if P and Q collide. (c) If λ = −1/2, find the shortest distance between P and Q.?

A particle P moving with a constant velocity i + j + k passes through a point with
position vector 3i −7j −4k at the same instant as a particle Q passes through a point
with position vector −i + j + λk. Q has a constant velocity vector 2i − j − 5k.

Answer 1

See below

Explanation:

(a)

Synchronising clocks:

`mathbf r_P = ((3),(-7),(-4)) + t ((1),(1),(1))`

`mathbf r_Q = ((-1),(1),(lambda)) + t ((2),(-1),(-5))`

`mathbf r_P = mathbf r_Q implies ((3),(-7),(-4)) + t ((1),(1),(1)) = ((-1),(1),(lambda)) + t ((2),(-1),(-5))`

From the first row: `3 + t = - 1 + 2t implies color(red)(t= 4)`.

Verify that by substituting `t = 4` into 2nd row. It follows from row 3 that if they collide: `lambda = 20`

(b)

The line that is perpendicular to both `mathbf r_P` and `mathbf r_Q` has direction:

`mathbf n = ((1),(1),(1)) times ((2),(-1),(-5))`

`= det ((hat i, hat j, hat k),(1,1,1),(2,-1,-5)) = -4hat i +7 hat j -3 hat k`

And unit vector: `mathbf hat n =1/sqrt(74) ((-4),(7),(-3))`

This is the direction of the shortest distance between `mathbf r_P` and `mathbf r_Q`.

We then project any vector connecting a point on each of `mathbf r_P` and `mathbf r_Q` onto that unit vector and find its magnitude:

`implies |(((3),(-7),(-4)) - ((-1),(1),(- 1/2)) ) * 1/sqrt(74) ((-4),(7),(-3)) |`

`= 123/(2 sqrt(74))`