Question

If the following lines are perpendicular to each other then λ equals : x-5/7=y-1/-5=z+2/1, x+2/λ=y+3/2=z+10/3 ?

Answer 1

`lambda = 1`

Explanation:

I'm guessing the slash is a fraction bar. Don't be shy about overparenthesizing folks, it helps us out here in answerland.

We'll use a parametric representation to shake out the direction vectors. The parameter is just the common ratio:

`(x-5)/7=(y-1)/-5 = (z+2)/1 = t`

`x =5+ 7t`

`y = 1 -5 t`

`z = -2 + t`

`(x,y,z) = (5,1,-2) + t(7, -5, 1)`

We see how the direction vector is just the denominators. Let's work out the other line:

`(x+2)/λ=(y+3)/2=(z+10)/3 =u`

`(x,y,z) = (-2,-3,-10) + u(lambda, 2, 3)`

We have perpendicularity when the dot product between the direction vectors is zero:

`0 = (7, -5, 1)cdot(lambda, 2, 3)= 7lambda -5(2)+1(3)`

`7 = 7 lambda`

`lambda = 1`

Check: `7(1)-5(2)+1(3)=0 quad sqrt`