Equation of a line perpendicular to 2 vectors

In summary: The cross product of two vectors, V1 and V2, is another vector that is perpendicular to both V1 and V2. It is denoted by V1 x V2. You can calculate it by taking the determinant of the matrix formed by the components of V1 and V2. The resulting vector will be perpendicular to both V1 and V2. In summary, to find a line in R3 passing through (-1,1,2) and perpendicular to both V1 (-1,1,-1) and V2 (1,1,1), you would need to calculate the cross product of V1 and V2, denoted by V1 x V2, and use it as the direction vector in the
  • #1
Bertrandkis
25
0

Homework Statement



Let L be a line in R3 passing through(-1,1,2) and is perpendicular to vectors V1 (-1,1,-1)
and V2 (1,1,1). Find an equation for L in parametric form.

Homework Equations





The Attempt at a Solution


using vector V2
(x,y,z)(1,1,1)=(1,1,1)(-1,1,2)
A possible equation for L=>(x,y,z)=(-1,1,2)+t(1,1,1)

I don't know if I am right.
 
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  • #2
Bertrandkis said:

Homework Statement



Let L be a line in R3 passing through(-1,1,2) and is perpendicular to vectors V1 (-1,1,-1)
and V2 (1,1,1). Find an equation for L in parametric form.

Homework Equations





The Attempt at a Solution


using vector V2
(x,y,z)(1,1,1)=(1,1,1)(-1,1,2)
A possible equation for L=>(x,y,z)=(-1,1,2)+t(1,1,1)

I don't know if I am right.
The line you give is not perpendicular to V2. Any point on (-1,1,2)+ t(1,1,1) has position vector (t-1, t+1, t+ 2) and its dot product with (1, 1, 1) is (t-1)+ (t+1)+ (t+2)= 3t+2, not 0. I'm not sure why you think that would give you a line perpendicular to (1,1,1).

But even if you were to find a line perpendicular to V2, there is no reason to think that line would also be perpendicular to V1! You need a direction vector that is perpendicular to both. Do you know how to calculate a vector perpendicular to both V1 and V2?
 
  • #3
To be honest, I don't know how to find a vector perpendicular to both V1 and V2.
I can find 2 equations of a lines perpendicular to each of the vectors
(1)=> (x,y,z)=(-1,1,2)+t(-1,1,1)
(2)=> (x,y,z)=(-1,1,2)+t(1,1,1)
now where do I go from here.
 
  • #4
Do you know what the cross product of two vectors is?
 

What is the equation of a line perpendicular to two vectors?

The equation of a line perpendicular to two vectors can be found by taking the cross product of the two vectors and setting it equal to the dot product of the perpendicular line and the two vectors.

How do you find the slope of a line perpendicular to two vectors?

The slope of a line perpendicular to two vectors can be found by taking the negative reciprocal of the slope of one of the vectors.

Can a line be perpendicular to two non-perpendicular vectors?

No, a line can only be perpendicular to two vectors if the two vectors are themselves perpendicular.

Can a line be parallel to two perpendicular vectors?

No, a line can only be parallel to two vectors if the two vectors are parallel to each other.

What happens if the two vectors are parallel?

If the two vectors are parallel, then their cross product will be equal to zero and there will be no unique equation for a line perpendicular to them.

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