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 **tail** to **tail**, move one vector so that it's **tail**
starts at the **tail** of the previous vector. Next, make a parallelogram. The **Resultant Vector** starts at the **tail**
of the vectors and ends at the opposite side of the parallelogram.

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

Notice that this method is very similar to the Triangle Law of Vector Addition. By moving vector
\(\vec{v}\) over to the head of \(\vec{u}\) makes a triangle. However, sometimes the exact direction of the vectors
is unknown. Instead, you may be provided information about the angle between the vectors. That is when the
**Parallelogram Law of Vector Addition** is most useful.

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

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

Show Answer