Unit Vectors

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.

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Definition 1.3.1. Unit Vectors.

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Symbols like \(\hat{x}\) and \(\hat{y}\) are special vectors called unit vectors or basis vectors. A unit vector has the following properties:

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A unit vector has a magnitude of 1
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A unit vector has no dimensions: the magnitude is 1, not 1 meter or 1 second
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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 \(x\)
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A unit vector is written with a “hat” symbol instead of an “arrow” symbol.
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You can turn any vector into a unit vector by dividing it by its own magnitude: for example, \(\hat{v} = \frac{\vec{v}}{|\vec{v}|}\)
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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.

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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).

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Since \(\hat{x} \text{,}\) \(\hat{y} \) and \(\hat{z} \) are mutually perpendicular unit vectors, their dot products are:

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\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.

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This short video includes a brief summary of unit vectors:

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