Direction of a Normal Vector

In summary, the normal vector for a plane with equation Ax + By + Cz + D = 0 is (A, B, C). However, it can also be written as (-A)x + (-B)y + (-C)z + (-D) = 0, in which case the normal vector is (-A, -B, -C) in the opposite direction. The direction of a normal vector is a matter of convenience and definition, with the common practice being to choose a unit vector (|z| = 1) and pointing "outwards".
  • #1
phiby
75
0
For a plane with equation Ax + By + Cz + D = 0, the normal vector is (A, B, C).

However, this plane equation can also be rewritten as (-A)x + (-B)y + (-C)z + (-D) = 0, in which case the normal vector is (-A, -B, -C) which is in the opposite direction as the other normal vector.

Basically my question is this - what's the direction of a normal vector - i.e. both directions seem correct.
 
Mathematics news on Phys.org
  • #2
Yes, if you simply think about the (x, y) plane, then (0, 0, 1 ) is a normal vector but so is (0, 0, -1). In fact, any vector of the form (0, 0, z) with z non-zero is a normal vector to the plane.
Which one you choose is a matter of convenience (usually |z| = 1 giving a unit normal vector is taken) and definition (if I define my z-axis to be the other way around, I can take the same normal vector but I will get a minus sign in the z-coordinate).

In some cases, for example on a sphere or other closed volume, it is common to pick the normal vector to that direction that our intuition calls "outwards", even though a vector pointing to the origin would also fit the definition.
 

What is a normal vector?

A normal vector is a vector that is perpendicular to a surface or a line. It is often used in mathematics and physics to describe the orientation or direction of an object or a force.

How is the direction of a normal vector determined?

The direction of a normal vector is determined by the right-hand rule. This means that if you curl your fingers in the direction of the normal vector, your thumb will point in the direction of the vector.

What is the importance of the direction of a normal vector?

The direction of a normal vector is important in many areas of science and engineering. For example, in geometry, it is used to calculate angles and determine the orientation of shapes. In physics, it is used to describe the direction of forces and in computer graphics, it is used to create realistic lighting effects.

How is the direction of a normal vector represented?

The direction of a normal vector is often represented using arrows or unit vectors. Arrows indicate the direction and length of the vector, while unit vectors have a magnitude of 1 and are used to simplify calculations.

Can the direction of a normal vector change?

Yes, the direction of a normal vector can change depending on the orientation of the surface or line it is perpendicular to. For example, if a surface is rotated, the direction of its normal vector will also change.

Similar threads

Replies
8
Views
966
Replies
7
Views
443
Replies
2
Views
1K
  • General Math
Replies
1
Views
647
Replies
1
Views
1K
Replies
5
Views
3K
Replies
3
Views
657
Replies
17
Views
2K
Replies
1
Views
239
Back
Top