Linear Algebra: Parallel, Perpendicular, or Neither?

[sdoyle]

Homework Statement

The line l passes through the point P=(1,-1,1) and has direction vector d=[2,3,-1]. Determine whether l and P are parallel, perpendicular, or neither to 2x+3y-z=1.

Homework Equations

n.p=n.x, cross product, dot product

The Attempt at a Solution

Would you just relate the direction vector, d, to the normal vector, n[2,3,-1] ?

 

[Dick]

Yes, you would just relate the normal vector to the direction vector. How do they relate?

 

[sdoyle]

they would be parallel because they are scalar multiples of one another (1 is the multiple) also the cross product would be zero indicating parallel vectors.

 

[Dick]

That means that the NORMAL to the plane is parallel to the line. So what's the relation between the line and the plane?

 

[sdoyle]

that they are perpendicular?

 

[Dick]

Yes, they are.

 

[sdoyle]

How would you prove that mathematically?

 

[Dick]

What's to prove? The normal vector of the plane is parallel to the line. You proved that by showing they are scalar multiples or using the cross product. You're done.