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14.4: Vector Normalization

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  • A vector of any given length can be divided by its modulus to create a unit vector (i.e. a vector of unit length). We will see applications of unit (or normalized) vectors in the next chapter.

    For example, the vector

    \[\mathbf{u}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+i\hat{\mathbf{k}} \nonumber\]

    has a magnitude:

    \[|\mathbf{u}|^2=1^2+1^2+(-i)(i)=3\rightarrow |\mathbf{u}|=\sqrt{3} \nonumber\]

    Therefore, to normalize this vector we divide all the components by its length:

    \[\hat{\mathbf{u}}=\frac{1}{\sqrt{3}}\hat{\mathbf{i}}+\frac{1}{\sqrt{3}}\hat{\mathbf{j}}+\frac{i}{\sqrt{3}}\hat{\mathbf{k}} \nonumber\]

    Notice that we use the “hat” to indicate that the vector has unit length.

    Need help? The links below contain solved examples.

    Operations with vectors:

    External links:

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