Question

Whta is the equation that has the same x-intercepts as L1:[x,y,z]= [3,5,-1] + t[4,-10,2] and is perpendicular to L2:[x,y,z]= [5,8,-2] + s[3,7,1]

Answer 1

V2 = 3 i + 7 j + 1 k

find vector perpendicular to that
V3 = Vx i + Vy j + Vz k
if perpendicular then
V3 * V2 = 0
3 Vx + 7 Vy + 1 Vz = 0
or
Vz = -(3Vx+7Vy)


now that vector has to go through the points where y and z are 0 for L1
if y = 0
t = 1/2
and L1 = [3+2 , 5-5 , -1 +1] = [5,0,0]
z is also 0 when t = 1/2
so
goes through [5,0,0 ]
L3 = [5,0,0] + p [1, 1, -10]

now let's check if
1 i + 1 j - 10 k is perpendicular to
3 i + 7 j + 1 k

dot product = 3 + 7 -10 = 0 sure enough