A **Position Vector** is a vector whose **tail** is moved to the origin. Remember that
vectors can change position (where they are in space) as long as they have the same magnitude and direction
(**Equal Vectors**).

Here, vector \( \vec{AB} = \vec{OP} \), i.e., they are **Equivalent Vectors**. Point \( O \) is the origin. It
is convienent to move vectors to the origin because if point is \( P(3,2) \) we know the x and y components so \( \vec{OP} = \left<3,2\right> \).

Otherwise, a position vector \( \vec{OP} \) can be found from points \( A(a_x,a_y) \) and \( B(b_x,b_y) \):

\( \vec{OP} = \vec{AB} = \left< b_x - a_x, b_y - a_y \right> \)

Vectors differ from lines because they have starting and end points and they can be moved around.

**Equivalent** or **Equal Vectors** have the same magnitude and direction but do not need to be in the same position:

\( \vec{u}=\vec{v}\)

**Negative** or **Opposite Vectors** have the same magnitude but opposite directions:

\( \vec{u}=-\vec{v}\)

\( |\vec{u}|=|\vec{v}|\)

If \(A\) and \(B\) are points in space then \(\vec{AB} =-\vec{BA} \)

The **Zero Vector** has a magnitude of 0 and no specified direction like \(\vec{0}\),
\( \textbf{0} \) or \( \left<0,0,0\right> \).

**Collinear** or **Parallel Vectors ** have the same direction but different magnitudes.
For example, \( \vec{u}=\left<1,1,1\right>\) and \( \vec{v}=\left<2,2,2\right>\). Since the vectors are in the same direction, we just need to stretch
(or compress) one out to make it equal to the other:

\( \vec{v}=\frac{1}{2}\vec{u}\)

Multiplying a vector by a scalar results in a new vector that has a different magnitude but same direction.

Vectors can also be parallel if they are in opposite directions by multiplying by a negative scalar.

The table below summarizes some properties of vectors:

\( \vec{u} + \vec{0} = \vec{u} \) |

\( \vec{u} + (-\vec{u}) = \vec{0} \) |

\( c (\vec{u} + \vec{v}) = c \vec{u} + c \vec{v} \) |

\( (c + d) \vec{u} = c \vec{u} + d \vec{u} \) |

\( (cd) \vec{u} = c (d \vec{u}) \) |

\( 1 \vec{u} = \vec{u} \) |