Question

The two lines with the equation r=(1,2,-44)+ t(k+1, 3k+1,k-3) and x=2-3s, y=1-10s, z=3-5s, are given.
a) determine the vaulue for K when the lines are paralel
b) determine the value of K when the lines are perpendictular

Answer 1

Vector 1 = (K+1) i +(3K+1)j + (K-3) k
Vector 2 = -3 i -10 j -5 k

If those are parallel the cross product is zero
i j k
K+1 3K+1 K-3
-3 -10 -5

i term -5(3K+1) + 10(K-3) = 0
K = -7
same for j and k directions, K = -7


If they are perpendicular the dot product is 0

(K+1)(-3) +(3K+1)(-10) + (K-3)(-5) = 0
solve for K