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

**Geometric** vectors can be subtracted 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. Subtracting vectors is the same as
adding the **Opposite** vector.

To subtract vectors **head** to **tail**, add the **Opposite** vector.
Move the **Opposite** vector so that it's **tail**
starts at the **head** of the positive vector. 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 subtract vectors \(\vec{x}\) and \(\vec{w}\) **head** to **tail**.
This method is referred to the **Parallelogram Law of Vector Addition** but we
are adding the **Opposite** vector.

The figure below shows how to add and subtract \(\vec{x}\) and \(\vec{w}\) :

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

A plane travels \( 200 [\frac{km}{hr}] N 20^\circ E \) when the wind speed is
\( 90 [\frac{km}{hr}] E \). What is the plane's velocity when there is no wind?

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To subtract vectors **tail** to **tail**, move the negative vector so that it's **tail**
starts at the **tail** of the positive vector. The **Resultant Vector** starts at the **head**
of the negative vector and ends at the **head** of the positive vector.

The figure below shows how to add and subtract \(\vec{x}\) and \(\vec{w}\) :

In parallelogram ABCDE, what is \(\vec{AB} - \vec{AD} \)?

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