Unit Vector VS Vector

• Unit Vector

 

A vector that has a magnitude (or length) of exactly 1. It is used to represent a direction without any information about magnitude or length.

 

$$\hat{n}=\left[\begin{array}{ccc}{\hat{n}}_x& {\hat{n}}_y& {\hat{n}}_z\end{array}\right]$$

 

- The components ${\widehat{n}}_x{\widehat{n}}_y{\widehat{n}}_z$ are the direction cosines, which are the projections of the unit vector along the x-axes, y-axes, and z-axes, respectively.

- The magnitude of the unit vector is always 1:
$|\widehat{n}|=\sqrt{{\widehat{n}}_x^2+{\widehat{n}}_y^2+{\widehat{n}}_z^2}=1$

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• Vector

 

It defined by both a direction and a magnitude. It can have any length, not necessarily 1.

 

$$\overrightarrow{u}=\left[\begin{array}{ccc}{u}_x& u_y& u_z\end{array}\right]$$

 

- The components $u_x{u}_y{u}_z$ describe the vector's projections along the 𝑥-axes, 𝑦-axes, and 𝑧-axes, respectively.

- It is generally not equal to 1, and it is calculated as:
$|\overrightarrow{u}|=\sqrt{u_x^2+u_y^2+u_z^2}$

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Example:

 

- Suppose you have a vector $\overrightarrow{v}=[2,3,6]$.

- The magnitude of this vector is:
$|\overrightarrow{v}|=\sqrt{2^2+3^2+6^2}=\sqrt{4+9+36}=\sqrt{49}=7$

- The unit vector in the same direction as can be found by dividing each component of by its magnitude:
$\widehat{v}=\frac{\overrightarrow{v}}{|\overrightarrow{v}|}=\frac{1}{7}[2,3,6]=\left[\frac{2}{7},\frac{3}{7},\frac{6}{7}\right]$