# The unit normal to a plane

• Supra
In summary, the task is to find the unit normal to the plane with the equation a + 2s - 2t = 15 and determine the distance of the plane from the origin. This can be solved using the formula s x t, where s and t are vectors, or by simplifying the equation to x + 2y - 2z = 15 and finding the normal using the formula r.n = p.

## Homework Statement

Find the unit normal to the plane a + 2s - 2t = 15. What is the distance of the plane from the origin?

## Homework Equations

The normal to a plane is given by s x t
For any plane, r.n = p [n = unit vector and p = constant]

## The Attempt at a Solution

Not entirely sure what I'm meant to be doing here as I'm not given any real values for the vectors s and t, so I can't see how to crossing them achieves anything. I'd be capable of crossing the two vectors if they were in component form, but not here. Obviously I'm missing something here, any help would be great.

Many thanks,
/Supra.

If a, s and t are vectors, that equation doesn't even make sense. The left side is a vector and the right side is a scalar. Are a, s and t the names of your coordinates?

My apologies, on reading the question again it seems the letters in the equation aren't meant to be vectors. So the equation is just a + 2s - 2t = 15 or to make it more simple: x + 2y - 2z = 15 where the letters are scalars I assume.

Supra said:
My apologies, on reading the question again it seems the letters in the equation aren't meant to be vectors. So the equation is just a + 2s - 2t = 15 or to make it more simple: x + 2y - 2z = 15 where the letters are scalars I assume.

That makes it easy, right? Now it's just your usual normal to a plane problem.

## 1. What is the definition of the unit normal to a plane?

The unit normal to a plane is a vector that is perpendicular to the plane and has a magnitude of 1. It is often denoted as n and is used to represent the orientation of the plane.

## 2. How is the unit normal to a plane calculated?

The unit normal to a plane can be calculated by finding the cross product of two non-parallel vectors that lie in the plane. The resulting vector will be perpendicular to both of these vectors and will have a magnitude of 1 when normalized.

## 3. Why is the unit normal to a plane important in mathematics and physics?

The unit normal to a plane is important because it is used to determine the angle of incidence and reflection of light or other electromagnetic waves. It is also used in calculating surface integrals and in vector calculus.

## 4. Can the unit normal to a plane be negative?

No, the unit normal to a plane can only have a positive magnitude of 1. However, its direction can be positive or negative, depending on the orientation of the plane.

## 5. How does the unit normal to a plane relate to the equation of the plane?

The unit normal to a plane is perpendicular to the plane and can be used to find the coefficients of the equation of the plane. The equation of a plane in vector form is n · (r - r0) = 0, where n is the unit normal, r is a position vector on the plane, and r0 is a known point on the plane.