Question

`vec(AB) = 3hati - hatj + hatk` and `vec(CD) = - 3hati + 2hatj + 4hatk` are two vectors. The position vectors of the points A and C are `6hati + 7hatj + 4hatk` and `-9hatj + 2hatk`, respectively. Find the position vector of a point P on the line AB and a point Q on the line CD such that `vec(PQ)` is perpendicular to `vec(AB)` and `vec(CD)` both.

Answer 1

We have, `vec(AB) = 3hati - hatj + hatk` and `vec(CD) = - 3hati + 2hatj + 4hatk` 

Also, the position vectors of A and C are `6hati + 7hatj + 4hatk` and `-9hatj + 2hatk`, respectively.

Since, `vec(PQ)` is perpendicular to both `vec(AB)` and `vec(CD)`

So, P and Q will be foot of perpendicular to both the lines through A and C.

Now, equation of the line through A and parallel to the vector `vec(AB)` is

`vecr = (6hati + 7hatj + 4hatk) + lambda(3hati - hatj + hatk)` 

And the line through C and parallel to the vector `vec(CD)` is given by

`vecr = - 9hatj + 2hatk + mu(-3hati + 2hatj + 4hatk)`  .....(i)

Let `vecr = (6hati + 7hatj + 4hatk) + lambda(3hati - hatj + hatk)` and `vecr = - 9hatj + 2hatk + mu(-3hati + 2hatj + 4hatk)`  .....(ii)

Let P(6 + 3λ, 7 − λ, 4 + λ) is any point on the first line and Q be any point on second line is given by 

(−3μ, −9 + 2μ, 2 + 4μ)

∴ `vec(PQ) = (-3mu - 6 - 3lambda)hati + (-9 + 2mu - 7 + lambda)hatj + (2 + 4mu - 4 - lambda)hatk`

= `(-3mu - 6 - 3lambda)hati + (2mu + lambda - 16)hatj + (4mu - lambda - 2)hatk`

If `vec(PQ)` is perpendicular to the first line, then

3(−3μ − 6 − 3λ) − (2μ + λ − 16) + (4μ − λ − 2) = 0

⇒ −9μ − 18 − 9λ − 2μ − λ + 16 + 4μ − λ − 2 = 0

⇒ −7μ − 11λ − 4 = 0    .....(iii)

If `vec(PQ)` is perpendicular to the second line, then −3(−3μ − 6 − 3λ) + (2μ + λ − 16) + (4μ − λ + 2) = 0

⇒ 9μ + 18 + 9λ + 4μ + 2λ − 32 + 16μ − 4λ − 8 = 0

⇒ 29μ + 7λ − 22 = 0  ......(iv)

On solving equations. (iii) and (iv)

We get −49μ − 77λ − 28 = 0

⇒ 319μ + 77λ − 242 = 0

⇒ 270μ − 270 = 0

⇒ μ = 1

Using μ in equation (iii), we get

−7(1) − 11λ − 4 = 0

⇒ −7 − 11λ − 4 = 0

⇒ −11 − 11λ = 0

⇒ λ = −1

∴ `vec(PQ) = [-3(1) - 6 - 3(-1)]hati + [2(1) + (-1) - 16]hatj + [4(1) - (-1) - 2]hatk`

= `-6hati - 15hatj + 3hatk`