14.4: Vector Normalization

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: http://tinyurl.com/mw4qmz8

External links:

• The dot product: http://patrickjmt.com/vectors-the-dot-product/
• The cross product: http://patrickjmt.com/the-cross-product/
• The dot and cross product: http://www.youtube.com/watch?v=enr7JqvehJs