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 ?

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Answer 1 · 2018-06-23T22:30:19

$lambda = 1$

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