Unit vector

Product of a scalar and a vector

The product of a scalar and a vector is a vector quantity. Graphically:

The magnitude of the resulting vector equals the product of the scalar and the magnitude of the original vector, and it’s parallel to it if the scalar is positive and opposite if the scalar is negative.

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Similarly we can divide a vector by a scalar; the magnitude of the resulting vector will be the magnitude of the original vector divided by the scalar:

We can now introduce the concept of unit vector.

Unit vector

A unit vector is a vector of length 1.

In Physics it’s useful to know how create a unit vector from another vector. This process is called normalization. A unit vector in created from another one by dividing the last by its magnitude:

This page will allow you to automatically calculate a unit vector from another vector.


There is a special type of unit vector (called the standard unit vectors) that are parallel to the coordinate axes, pointing towards positive values of the coordinates. For each of the three coordinate axes x, y, z these vectors are called respectively i, j y k (or ux, uy y uz):

The standard unit vectors can be rotated depending on the orientation of the coordinate system, but they must follow a certain order. In the figure bellow a right-handed coordinate system is shown:

Unit vector notation

In the fields of Physics and Engineering vectors are often expressed in terms of the three unit vectors i, j, k.

In the figure below a 2D vector is shown:

As we said when we explained the scalar and vector quantities, its components are given by:

But vector a can also be written as the sum of ax and ay, respectively in blue and green in the figure:

ax and ay are the projections of a along the axes multiplied by the corresponding standard unit vector:

And therefore vector a will be given by:

This way of expressing a vector is called unit vector notation.

In three dimensions, we have to add the z component of a, az, multiplied by k.

From now on we’ll use this notation to write vectors.

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