Unit vector orthogonal to plane

In summary, the conversation discusses finding a unit vector with a positive first coordinate that is orthogonal to a plane passing through three given points. The solution involves computing the cross product of two vectors and ensuring that the resulting vector is a unit vector.
  • #1
olivia333
12
0

Homework Statement



Find a unit vector with positive first coordinate that is orthogonal to the plane through the points P = (-4, 5, 4), Q = (-1, 8, 7), and R = (-1, 8, 8).

Homework Equations



u = PQ = Q - P
v = PR = R - P
ans = uXv = PQ X PR

The Attempt at a Solution



so I did:
PQ = Q - P = (3,3,3)
PR = R - P = (3,3,4)

Then I computed the cross product to get the answer.

|i..j..k|
|3 3 3| = <(12-9) , -(12-9) , (9-9)> = <3, -3, 0>
|3 3 4|

The 0 is correct but the 3 and -3 are not. What am I doing wrong?
Thanks!
 
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  • #2
welcome to pf!

hi olivia! welcome to pf! :smile:
olivia333 said:
The 0 is correct but the 3 and -3 are not. What am I doing wrong?

not a unit vector? :wink:
 
  • #3
ohhhh haha thank you so much!
 

What is a unit vector orthogonal to a plane?

A unit vector orthogonal to a plane is a vector that is perpendicular to all the vectors in the plane. This means that it forms a 90 degree angle with every vector in the plane.

How do you find a unit vector orthogonal to a plane?

To find a unit vector orthogonal to a plane, you can use the cross product of two non-parallel vectors in the plane. The resulting vector will be orthogonal to the plane and then you can divide it by its magnitude to get a unit vector.

Why is a unit vector orthogonal to a plane important?

A unit vector orthogonal to a plane is important because it helps us understand the orientation of the plane in three-dimensional space. It also allows us to calculate the distance of a point from the plane, which is useful in many applications.

What is the relationship between a unit vector orthogonal to a plane and the normal vector of the plane?

The unit vector orthogonal to a plane is the same as the normal vector of the plane, but with a magnitude of 1. This means that it has the same direction as the normal vector, but is scaled down to a unit vector.

Can there be more than one unit vector orthogonal to a plane?

Yes, there can be infinitely many unit vectors orthogonal to a plane. This is because any vector that is perpendicular to the plane and has a magnitude of 1 can be considered a unit vector orthogonal to the plane.

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