Unit vectors: characteristics, how to get it, examples

Recall that vectors are mathematical entities that mathematically represent physical magnitudes that depend on direction, such as force, velocity, acceleration, and others.unit vectors

Regardless of the physical magnitude to which they are associated, unit vectors are devoid of units of measurement and their size is always 1, a pure number.

For example, the velocity of a particle moving at 3 m / s and going in the positive direction of the Cartesian axis X is denoted: v = (3 m / s) i , where the bold letter is used to denote vector quantities. In this example the modulus of v is 3 m / s and the modulus of the unit vector i is 1 (unitless).

Module, direction and sense

Given how important it is to establish the orientation of these quantities in order to know their effects, vectors have three relevant characteristics: the magnitude or module, associated with the size of the vector, the direction and the sense. When representing a vector quantity it is necessary to clearly indicate these aspects.

Now, a unit vector can have any direction and the sense that is preferred, but the magnitude must always be equal to 1.

Unit vectors are used to point to a particular direction in space or in the plane. If, for example, we need to work with all the forces that act along the horizontal axis, then a unit vector in that direction helps us to distinguish these forces from others directed in a different direction.

And to distinguish them from non-unit vectors, bold type is usually used in printed letter and a caret is placed on top, for example:

Characteristics of a unit vector

Mathematically the unit vector:

u o v

Then we can establish that:

-The module of the unit vector is always 1, it does not matter if it is a force, velocity or other vector.

-The unit vectors have a certain direction, as well as sense, such as the unit vector in the vertical direction, which can have sense up or down.

-Unit vectors have a point of origin. When represented by a Cartesian coordinate system, this point coincides with the origin of the system: (0,0) if it is the plane or (0,0,0) if the vector is in three-dimensional space.

-Also with unit vectors you can perform all vector addition, subtraction and multiplication operations that are done using regular vectors. Therefore, it is valid to multiply the unit vector by a scalar, as well as to carry out the point product and the cross product.

-With a unit vector in a certain direction, other vectors that are also oriented in that direction can be expressed.

Unit vectors in space

To express any vector in space or in the plane, a set of unit vectors perpendicular to each other can be used, which form an orthonormal basis. Each of the three preferential directions of space has its own unit vector.

Let’s go back to the example of forces directed along the horizontal axis. This is the x-axis, which has two possibilities: to the right and to the left. Suppose we have a unit vector on the x-axis and directed to the right, which we can denote by any of these ways:

Any one of them is valid. Now, suppose a force F ₁ of magnitude 5 N along this axis and directed to the right, such a force could be expressed as:

If the force were directed along the x-axis but in the opposite direction, that is, to the left, then a negative sign could be used to establish this difference.

For example, a force of magnitude 8 N, located on the x-axis and directed to the left, would look like this:

Or like this:

And for the vectors that are not directed along the Cartesian axes, there is also a way to represent them in terms of the orthogonal unit vectors, by their Cartesian components.

How to get / calculate the unit vector?

To calculate the unit vector in the direction of any arbitrary vector v , the following formula is applied:

Where:

It is the module or magnitude of the vector v , whose square is calculated as follows:

| v | ² = (v _x ) ² +  (v _y ) ² + ( v _z ) ²

An arbitrary vector in terms of the unit vector

Alternatively the vector v can be expressed like this:

That is, the product of its module by the corresponding unit vector. This is exactly what was done earlier, when talking about the force of magnitude 5 N directed along the positive x axis.

Graphic representation

Graphically, the aforementioned is seen in this image, where the vector v is in blue and the corresponding unit vector in its direction is in red. unit vectors

In this example, the vector v has a magnitude greater than that of the unit vector, but the explanation is valid even if it does not. In other words, we can have vectors that are for example 0.25 times the unit vector.

Examples of unit vectors

The perpendicular unit vectors i, j, and k

As we have seen before, the perpendicular unit vectors i , j and k are very useful to represent any other vector in the plane or space, and to carry out vector operations. In terms of these vectors, an arbitrary vector v is represented as:

v = v _x i + v _y j + v _z k

Where v _x , v _{y and} v _z are the rectangular components of the vector v , which are scalars – no bold type is used to represent them in printed text.

Coulomb’s law unit vectors

Unit vectors appear frequently in Physics. There we have Coulomb’s law, for example, which quantitatively describes the interaction between two point electric charges.

It states that the force F of attraction or repulsion between said charges is proportional to their product, inversely proportional to the square of the distance that separates them and is directed in the direction of the unit vector that joins the charges.

This vector is usually represented by:

And Coulomb’s law looks like this, in vector form:

Exercise resolved

Find the unit vector in the direction of the vector v = 5 i + 4 j -8 k , given in arbitrary units.

Solution unit vectors

The definition of unit vector given above applies:

But first, we must calculate the module of the vector, which as it has three components, is determined by:

| v | ² = (v _x ) ² +  (v _y ) ² +  (v _z ) ²

Remaining:

| v | ² = (5) ² +  (4) ² +  (-8) ² = 25 + 16 + 64 = 105

Therefore the modulus of v  is:

| v | = √105

The unit vector searched is simply:

Which finally leads us to:

 v = 0.488 i + 0.390 j – 0.781 k