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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 1.3 Unit Vectors

You will often need to orient a vector within a particular coordinate system. A unit vector is a vector that has magnitude of one. For this reason, unit vectors are sometimes called direction vectors.

Definition 1.3.1. Unit Vectors.

Symbols like \(\hat{x}\) and \(\hat{y}\) are special vectors called unit vectors or basis vectors. A unit vector has the following properties:
  • A unit vector has a magnitude of 1
  • A unit vector has no dimensions: the magnitude is 1, not 1 meter or 1 second
  • A unit vector always points in the direction along which the given variable is increasing: for example, \(\hat{x}\) points in the direction of increasing \(x\)
  • A unit vector is written with a “hat” symbol instead of an “arrow” symbol.
  • You can turn any vector into a unit vector by dividing it by its own magnitude: for example, \(\hat{v} = \frac{\vec{v}}{|\vec{v}|}\)
Some authors, textbooks, and resources will use \(\hat{i}\) and \(\hat{j}\text{,}\) respectively, instead of \(\hat{x}\) and \(\hat{y}\text{,}\) but they mean the same thing. You can learn more in Unit Vectors.
You represent a unit vector symbolically with a “hat”. Most commonly, unit vectors are used for indicating the directions of a coordinate system. For example \(\hat{x} \text{,}\) \(\hat{y} \) and \(\hat{z} \) are unit vectors which point parallel to the positive \(x\)-direction, positive \(y\)-direction and positive \(z\)-direction (these are often called basis vectors in a mathematics class).
Since \(\hat{x} \text{,}\) \(\hat{y} \) and \(\hat{z} \) are mutually perpendicular unit vectors, their dot products are:
\begin{equation*} \hat{x} \cdot \hat{x} = \hat{y} \cdot \hat{y} = \hat{z} \cdot \hat{z} = 1 \end{equation*}
\begin{equation*} \hat{x} \cdot \hat{y} = \hat{x} \cdot \hat{z} = \hat{y} \cdot \hat{z} = 0 \end{equation*}

Note 1.3.2. Unit Vector Notation.

Different textbooks and authors use different notation for the unit vectors. Often, you will see unit vectors written using \(\hat{i} \text{,}\) \(\hat{j} \) and \(\hat{k} \) to represent the positive \(x\)-direction, positive \(y\)-direction, and positive \(z\)-direction, respectively. Both notations are equivalent, but many physicists use \(\hat{x} \) instead of \(\hat{i} \) because it more directly evokes the direction in which the unit vector points.
This short video includes a brief summary of unit vectors:

Exercises Practice Activities

1.

Sketch a unit vector that points in the same direction as the vector \(\vec{v_1}= 3\hat{x}-4\hat{y}\text{.}\)

2.

Sketch a unit vector that points perpendicular to the vector \(\vec{v_2}= -12\hat{x}-5\hat{y}\text{.}\)

References References

[1]
Coordinate systems and unit vectors by Dr. Michelle Tomasik from MIT 8.01 Classical Mechanics, Fall 2016, used under Creative Commons BY-NC-SA.
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