What are the implications of $\vec{AB} = \vec{CD}$?

Question

$$\vec{AB} = \vec{CD}$$

For a moment, I had thought that

$$A = C \\ B = D$$

Which is not necessarily true - this was a little counter-intuitive for me, because I see vectors $\vec{XY}$ as "arrows with position", and thus it seemed natural for me that two vectors $\vec{AB}$ and $\vec{CD}$ should have the same position to be equal - which is not true. So I was wondering if there is another property related to equality that I may be misinformed of. Therefore:

What are the implications of

$$\vec{AB} = \vec{CD}$$

?

As far as I am concerned:

• $\vec{AB}$ is parallel to $\vec{CD}$
• The distance between $A$ and $B$ is the same as from $C$ to $D$.

Answer 1

$\|\vec{AB}\|=\|\vec{CD}\|$. Also, the directions of the two vectors are the same, i.e., they are parallel, not antiparallel. All their corresponding components are equal.

Answer 2

We can distinguish 'directed segments' and 'vectors': let vectors be the directed segments, but let the equality for them defined so that if we shift a directed segment parallelly to anywhere else, it should be equal to the origin one, as a vector.

By the way, we have $\vec{AB}\ =\ \vec{CD}\ $ iff $ $ the quadrilateral $ABDC$ is a parallelogram (with this orientation).

Answer 3

If the points are given coordinates, then $\vec{AB} = \vec{CD}$ is the same as $B - A = D - C$.

Answer 4

I believe you have all of the conditions for equality. It is counter intuitive at first that vectors at different positions are equal, but this makes sense after you manipulate them in various applications. For instance you may define an acceleration or velocity by a vector quantity, in this regard it makes sense for a push north of 2 newtons to be equal to any other push north by 2 newtons, or going 60 kph east on your favorite highway is the same as going 60 kph east anywhere.

Answer 5

$$\left(\vec{AB} = \vec{CD}\right) \iff \left(\overline{AB} \cong \overline{CD}\right) \bigwedge \left(\vec{AB} \parallel \vec{CD}\right)$$

Vectors are considered equal if their line segments are congruent and they are parallel.  That is, they have the same size and direction.

The position of the line segment $\overline{AB}$ is not considered to be a property belonging to the vector $\vec{AB}$.  The vector describes relative displacement between the points, rather than absolute placement of the points.

Note: Orientation does matter.  Vectors traversing along exactly opposing directions are considered to be antiparallel.