# Statics: Dimensionless Unit Vector

• belvol16
In summary, the conversation discusses finding the unit vector in a positive direction by dividing the vector by its magnitude. The online book lists the answer as a unit vector with components of 0.0720 i and 0.997 j. The purpose of unit vectors is to determine the direction of the vector, and they always have a magnitude of 1. However, there is confusion about finding a dimensionless unit vector in this direction.
belvol16

## The Attempt at a Solution

So I began by subtracting.
(205-160)=55 i
(495+128)=623 j
Both of these vectors are in the positive direction. So if I divide the vector by its magnitude I should get an answer of 1 in the positive direction for both i and j.
The online book lists the answers as R hat = 0.0720 i + 0.997 j . I guess I'm not sure how the book got these answers. I thought unit vectors always had a magnitude of 1 and their purpose was to determine the direction of the vector.
Thank You!

belvol16 said:
So if I divide the vector by its magnitude I should get an answer of 1 in the positive direction for both i and j.
If each component had a magnitude of 1 then the vector would not be a unit vector!

belvol16 said:
The online book lists the answers as R hat = 0.0720 i + 0.997 j . I guess I'm not sure how the book got these answers. I thought unit vectors always had a magnitude of 1 and their purpose was to determine the direction of the vector.
The unit vector does have a magnitude of 1. Given those components, find the magnitude of the full vector. What do you get?

Minor point, but ##\vec{B} - \vec{A}## has units of lbs, and so would a unit vector in this direction. I'm puzzled by the instruction to "find a dimensionless unit vector" in this direction.

Mark44 said:
Minor point, but ##\vec{B} - \vec{A}## has units of lbs, and so would a unit vector in this direction. I'm puzzled by the instruction to "find a dimensionless unit vector" in this direction.
If you divide by the magnitude (also in lbs) you can get a dimensionless direction vector.

Doc Al said:
If you divide by the magnitude (also in lbs) you can get a dimensionless direction vector.
OK, that makes sense. Objection withdrawn...

## 1. What is a dimensionless unit vector?

A dimensionless unit vector is a vector that has a magnitude of 1 and has no units. It represents a direction in space without any reference to a specific unit of measurement.

## 2. How is a dimensionless unit vector different from a regular unit vector?

A regular unit vector also has a magnitude of 1, but it is associated with a specific unit of measurement. A dimensionless unit vector, on the other hand, has no units and represents a direction in space without any reference to a specific unit of measurement.

## 3. Why are dimensionless unit vectors important in statics?

In statics, dimensionless unit vectors are important because they allow us to describe the direction of a force or moment without being restricted to a specific unit of measurement. This makes calculations and equations more general and applicable to a wider range of situations.

## 4. How do you calculate a dimensionless unit vector?

To calculate a dimensionless unit vector, you first need to find the magnitude of the vector using the Pythagorean theorem. Then, divide each component of the vector by its magnitude to get the unit vector. Finally, remove any units from the resulting vector to make it dimensionless.

## 5. Can you have a dimensionless unit vector with negative components?

Yes, it is possible to have a dimensionless unit vector with negative components. The magnitude of the vector will still be 1, but the direction will be in the opposite direction of the negative component. For example, if a vector has a component of -1 in the x-direction, the resulting dimensionless unit vector will have a component of 1 in the negative x-direction.

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