# Unit Vectors

*is a vector with magnitude of 1. In some situations it is helpful to find a unit vector that has the same direction as a given vector.*

__unit vector__A unit vector of

**v**, in the same direction as

**v**, can be found by dividing

**v**by its magnitude $\parallel v\parallel $.

UNIT VECTOR:

*If $v\ne 0$ and $\parallel v\parallel $ represents the magnitude of vector v, then its unit vector u is:*

$u=\frac{v}{\parallel v\parallel}=\frac{1}{\parallel v\parallel}v$

*The unit vector u has a length of 1 in the same direction as v.*

The unit vectors $\langle 1,0\rangle $ and $\langle 0,1\rangle $ are special unit vectors called

*and are represented by the vectors i and j as follows:*

__standard unit vectors__$i=\langle 1,0\rangle $ $j=\langle 0,1\rangle $

Any vector in a plane can be written using these standard unit vectors.This vector sum is called a

*. For example, vector $v=\langle 3,11\rangle =3i+11j$.*

__linear combination__Let's look at some examples.

*To work these examples requires the use of various differentiation rules. If you are not familiar with a rule go to the associated topic for a review.*

**Example 1: Find a unit vector u in the same direction as $v=\langle 12,-9\rangle $ and show that it has a magnitude of 1.**

The magnitude of a vector is calculated by taking the square root of the sum of the squares of its components. $v=\langle x,y\rangle $ $\parallel v\parallel =\sqrt{{x}^{2}+{y}^{2}}$ |
$v=\langle 12,-9\rangle $ ||v|| = $\sqrt{{12}^{2}+{\left(-9\right)}^{2}}$ $\parallel v\parallel =\sqrt{225}={15}$ |

$u=\frac{v}{\parallel v\parallel}=\frac{1}{\parallel v\parallel}v$ |
$u=\frac{v}{\parallel v\parallel}=\frac{1}{\parallel v\parallel}v$ $u=\frac{\langle 12,-9\rangle}{15}=\frac{1}{15}\langle 12,-9\rangle $ $u=\langle \frac{12}{15},-\frac{9}{15}\rangle ={\langle}\frac{4}{5}{,}{-}\frac{3}{5}{\rangle}$ |

The magnitude of a vector is calculated by taking the square root of the sum of the squares of its components. $v=\langle x,y\rangle $ $\parallel v\parallel =\sqrt{{x}^{2}+{y}^{2}}$ |
$u=\langle \frac{4}{5},-\frac{3}{5}\rangle $ ||u|| = $\sqrt{{\left(\frac{4}{5}\right)}^{2}+{\left(-\frac{3}{5}\right)}^{2}}$ ||u|| = $\sqrt{\frac{16}{25}+\frac{9}{25}}=\sqrt{\frac{25}{25}}$ $\parallel u\parallel =\sqrt{1}=1$ |

**Example 2: If u = -3i + 2j and v = -i + 6j, find 2u + 4v.**

$kv=k{v}_{1},{v}_{2}=k{v}_{1},k{v}_{2}\to ScalarMultiplication$ |
$2u=2\left(-3i+2j\right)=\left[2\xb7\left(-3i\right)+2\xb72j\right]$ $2u=-6i+4j$ $4v=4\left(-i+6j\right)=\left[4\xb7\left(-i\right)+4\xb76j\right]$ $4v=-4i+24j$ |

$u+v={u}_{1}+{v}_{1},{u}_{2}+{v}_{2}\to VectorAddition$ |
2u + 4v (-6i + 4j) + (-4i + 24j) (-6i - 4i) + (4j + 24j) $\left(-10i+28j\right)$ |

Related Links:Math algebra Direction Angles of Vectors The Dot Product Pre Calculus |

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