1. ## perpendicular vectors

I need to evaluate if two lines are perpendicular and give a reason for my answer.

So: To show that the lines are not perpendicular, I need to show by the dot product method that my calculation does not = 0. Because cos(0)=90, perpendicular lines.
I have two lines below:
I need to start with my components in (i, j, k) form.
The angle between the lines is the angle between their directions. The directions are U and V.

r1=j+ak +λ(2i+4j-3k)
r2=4i+3j-6k+μ(i+5j+k)

u=2i+4j-3k,
v=i+5j+k,
u.v=(2, 4, -3)(1, 5, 1)=(2+20-3)=19

Is this enough or do I need to go further, i.e

|u|=√2+4+3=3
|v|=√2+20+3=5

cosθ=u.v/|u|.|v|
19/15=1.267

θ=but when I inverse cos I get a mathematical error when I try to inverse cos this figure?

2. If two lines do not intersect, can they be perpendicular?

3. no

4. Therefore, if the dot product of the vectors is 0, it is not necessarily true that the lines are perpendicular (in 3-D).

5. ahh shoot! That was dumb. sorry
its not:
|u|=√2+4+3=3
|v|=√2+20+3=5

but
|u|=√2²+4²+3²=√29
|v|=√2²+20²+3²=√413

cosθ=u.v/|u|.|v|
19/√29.√413=1.17
θ=80°
Therefore the lines are not perpendicular.