Unit Vector

A unit vector is a vector whose magnitude is exactly 1. It shows direction only, without any scale or length attached to it.

If a vector \(\vec{v}\) points in some direction, its corresponding unit vector points the same way but has length 1.

Unit vectors are useful when you want to express direction independently of magnitude.

To convert any vector into a unit vector, you divide the vector by its magnitude.

If \(\vec{v}\) is any non-zero vector, the unit vector in its direction is:

\[ \hat{v} = \dfrac{\vec{v}}{|\vec{v}|} \]

This makes the magnitude of \(\hat{v}\) equal to 1.

For a vector in 2D, \(\vec{v} = \langle a, b \rangle\), the unit vector is:

\[ \hat{v} = \left\langle \dfrac{a}{\sqrt{a^2 + b^2}},\; \dfrac{b}{\sqrt{a^2 + b^2}} \right\rangle \]

For a vector in 3D, \(\vec{v} = \langle a, b, c \rangle\), the formula becomes:

\[ \hat{v} = \left\langle \dfrac{a}{\sqrt{a^2 + b^2 + c^2}},\; \dfrac{b}{\sqrt{a^2 + b^2 + c^2}},\; \dfrac{c}{\sqrt{a^2 + b^2 + c^2}} \right\rangle \]

If you multiply a unit vector by a positive number, it keeps its direction but the magnitude changes.

If you multiply it by a negative number, the direction reverses, but the vector still follows the same line.

Example 1:

For \(\vec{v} = \langle 6, 8 \rangle\), magnitude is:

\[ |\vec{v}| = \sqrt{6^2 + 8^2} = 10 \]

The unit vector is:

\[ \hat{v} = \left\langle \dfrac{6}{10}, \dfrac{8}{10} \right\rangle = \langle 0.6, 0.8 \rangle \]

Example 2:

For \(\vec{v} = \langle 2, -1, 2 \rangle\), magnitude is:

\[ |\vec{v}| = \sqrt{2^2 + (-1)^2 + 2^2} = 3 \]

Unit vector:

\[ \hat{v} = \left\langle \dfrac{2}{3}, -\dfrac{1}{3}, \dfrac{2}{3} \right\rangle \]