Vector application problem? ?

Question

Can two vectors be perpendicular to each other if they both start at the same origin and have opposite x-components? How would I prove this right or wrong. I don't get how to represent the vectors in this situation. What would these vectors look like?

Answer 1

Yes the can.

A simple set having the desired properties would be

v₁ = (1,1,0)  and v₂(-1,1,0)

If you sketch the two vectors in the xy-plane, you will see that I'm right.

To prove it formally that such a set of vectors exist use the the dot product. The dot product of two perpendicular vectors equals zero.

Let

v₁ = (x₁,y₂,z₂)  and v₂(x₂,y₂,z₂) be two vectors.

They are perpendicular so:

v₁∙ v₂ = 0

<=>

x₁∙x₂ + y₁∙y₂ + z₁∙z₂ = 0

The vectors have opposite x-components

x₂ = -x₁

Hence:

-x₁² + y₁∙y₂ + z₁∙z₂ = 0

<=>

y₁∙y₂ + z₁∙z₂ = x₁²

There are many sets of vectors satisfying this condition. Since you have one equation for five unknowns you can choose arbitrary values for four components and then calculate the last one from the equation

For the set above i choose

x₁ = 1

z₁ = z₂ = 0

and

y₁ = 1

=>

y₂ = (x₁² - z₁∙z₂) / y₁ = 1

Many other sets exist but it's sufficient for the proof of the statemant that one exists.

Answer 2

Yes, these vectors can be perpendicular.  They look like an arrow /\ or\/.  a 45degree angle and a 135 degree angle have opposite x components, are perpendicular to each other, and can have the same origin.  Same goes for 225 and 315

Answer 3

vecA don=vec0

vecBdont=vec0

then vecA.vecB=0 IF and only if[ vecA perpendiclar vecB]