Adding vectors results in a new **Resultant Vector**. **Geometric** and **algebraic** vectors can be added using different techniques.

**Geometric** vectors can be added by positioning the vectors **head** to
**tail** or **tail** to **tail**. Remember, you can move a vector around
in space as long as it maintains its magnitude and direction.

To add vectors **head** to **tail**, move one vector so that it's **tail**
starts at the **head** of the previous vector. Repeat for all vectors being added teogether. The **Resultant Vector** starts at the **tail**
of the first vector and ends at the **head** of the last one.

Below are the steps to add vectors \(\vec{u}\) and \(\vec{v}\) **head** to **tail**.
Sometimes, this method is referred to the **Triangle Law of Vector Addition**.

These steps can be extended to more than one vector.

Notice that it doesn't matter which order you add the vectors in. These rules are shown below:

Commutative Law of Addition | \(\vec{u} + \vec{v} = \vec{v} + \vec{u}\) |

Associative Law of Addition | \( (\vec{m} + \vec{n}) + \vec{o} = \vec{m} + (\vec{n} + \vec{o})\) |

Adding **geometric** vectors involves solving triangles using trigonometry equations.
Below is a summary:

Right Triangles | |

Pythagorean Theorem | \(c^2 = a^2 + b^2 \) |

SOH CAH TOA |
\(sin(\theta) = \frac{opp}{hyp} = \frac{y}{r}\) \(cos(\theta) = \frac{adj}{hyp} = \frac{x}{r} \) \(tan(\theta) = \frac{opp}{adj} = \frac{y}{x}\) |

Other Triangles | |

Cosine Law | \(c^2 = a^2 + b^2 - 2 a b cos(\theta)\) |

Sin Law | \(\frac{sin(A)}{a} = \frac{sin(B)}{b} = \frac{sin(C)}{c} \) |

While driving, Zaina travels \( 5 \; [km] \; N \) and then \( 2 \; [km] \; E \).
Find her displacement.

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Frank and Michelle take their son Marco tobogganing. On the way to
the slopes, Macro sits on the toboggan and Frank and Michelle each pull on a string attached to the front.
Frank applies a force of 45 [N] \(30^\circ\) away from the toboggan. Michelle applies a force of 26 [N] \( 60^\circ\)
away from the toboggan in the other direction. What is the resultant force applied to Marco on the toboggan?

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Frank and Michelle take their son Marco tobogganing. On the way to
the slopes, Macro sits on the toboggan and Frank and Michelle each pull on a string attached to the front.
Frank applies a force of 45 [N] \(30^\circ\) away from the toboggan. Michelle applies a force of 26 [N] \( 30^\circ\)
away from the toboggan in the other direction. What is the resultant force applied to Marco on the toboggan?

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Supose you add \(\vec{AB} + \vec{BC} \). Adding the vectors **head** to **tail**
shows that the result is \(\vec{AC} \). When the middle letters match, you can "collapse" them to quickly
find the **Resultant Vector**.

In pentagon ABCDE, what is \(\vec{AB} + \vec{DE} + \vec{BD} \)?

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