Find the distance of the point (−1, −5, −10) from the point of intersection of the line `vecr=2hati-hatj+2hatk+lambda(3hati+4hatj+2hatk) ` and the plane `vec r (hati-hatj+hatk)=5`

#### Solution

Let us suppose that the given line and plane intersect at the point P(x,y,z).

∴ The position vector of P is `vecr=xhati+yhatj+zhatk`

Thus, the given equations of the line and the plane can be rewritten as

`xhati+yhatj+zhatk=(2+3λ)hati−(1+4λ)hayj+(2+2λ)hatk and (xhati+yhatj+zhatk).(hati−hatj+hatk) = 5,` respectively.

On simplifying `xhati+yhatj+zhatk=(2+3λ)hati−(1+4λ)hatj+(2+2λ)hatk and (xhati+yhatj+zhatk).(hati−hatj+hatk) = 5` , we get:

x=(2+3λ)y =−(1+4λ)z = (2+2λ)

Also, x−y+z = 5

On putting the values of *x*, *y* and *z* in the equation x−y+z = 5, we get:

2+3λ+1+4λ+2+2λ = 5⇒9λ+5 = 5⇒λ = 0

∴ x=2, y =−1 and z =2

Hence, the distance between the points (−1, −5, −10) and (2, −1, 2) is

`sqrt((2+1)^2+(−1+5)^2+(2+10)^2)=sqrt(9+16+144)=sqrt169=13 units`