# perpendicular vectors

• Feb 20th 2010, 11:59 AM
stealthmaths
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?
• Feb 20th 2010, 12:06 PM
icemanfan
If two lines do not intersect, can they be perpendicular?
• Feb 20th 2010, 12:32 PM
stealthmaths
no
• Feb 20th 2010, 01:03 PM
icemanfan
Therefore, if the dot product of the vectors is 0, it is not necessarily true that the lines are perpendicular (in 3-D).
• Feb 20th 2010, 04:44 PM
stealthmaths
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.