Math Questions and Answers

Start Your Free Trial

Determine if the line `(x-1)/2`=`(y-2)/3`=`(z+3)/-2` and `(x-1)/-1`=`(y-2)/2`=`(z+3)/2` are perpendicular.  

Expert Answers info

Luca B. eNotes educator | Certified Educator

calendarEducator since 2011

write5,348 answers

starTop subjects are Math, Science, and Business

You need to convert the symmetric forms of equation of first line in parametric form such that:

`(x-1)/2 = t =gt x = 1 + 2t`

`(y-2)/3 = t =gt y = 2 + 3t`

`(z+3)/(-2) = t =gt z = -3 - 2t`

Hence, the parametric equations of the first line are `x = 1 + 2t ; y = 2 + 3t ;z = -3 - 2t`  and the direction vector of the line is `bar v_1 = lt2,3,-2gt.`

You need to convert the symmetric forms of equation of second line in parametric form such that:

`(x-1)/(-1) =s =gt x = 1- s`

`(y-2)/2 =s =gt y = 2 + 2s`

`(z+3)/2 = s =gt z = -3 - 2s`

The direction vector of the second line is `barv_2 = lt-1,2,-2gt`

You need to remember that these two lines are orthogonal if the cross product of direction vectors is zero, hence you should check if `lt2,3,-2gt*lt-1,2,-2gt = 0` .

`barv_1*barv_1 = 0`

`lt2,3,-2gt*lt-1,2,-2gt = 2*(-1) + 3*2 + (-2)*(-2)`

`lt2,3,-2gt*lt-1,2,-2gt = -2 + 6 - 4`

`lt2,3,-2gt*lt-1,2,-2gt = 0`

Hence, since `barv_1*barv_1 = 0`  , then the lines are orthogonal.

check Approved by eNotes Editorial