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Unit vectors
Beginner
Vectors of length one giving direction: i, j, k.
A = Axî + Ayĵ + Azk̂Unit vectors have length 1 and show direction only: î, ĵ, k̂ along x, y, z. Any vector is A = Axî + Ayĵ + Azk̂. Dividing a vector by its magnitude gives its unit vector.
The vector (3, 4) has magnitude 5 and unit vector (0.6, 0.8).
The hat tells you it carries no units; the magnitude carries the units.
In more detail
A unit vector has magnitude one and points in a given direction. Unit vectors i, j and k point along the x, y and z axes and make it easy to write any vector in component form.
Real-world uses
- Writing the position of a drone as r = 3i + 4j in computer graphics and robotics.
- Expressing directions in navigation and 3D modelling software.
- GPS navigation software uses unit vectors to describe the direction of a path at each point, separate from its length.
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Frequently asked questions
How do I find a unit vector in the direction of A?
Divide A by its magnitude, giving A divided by |A|.
Are unit vectors dimensionless?
Yes. They have magnitude one, so they carry direction without units.
Why do we use unit vectors?
They separate direction from magnitude, so a vector can be written as a number times a direction, which keeps calculations clear.
Related topics in Measurement & Math Tools
Vector addition · Parallelogram law · Vector components · Dot product · Cross product · Triple products