# What is the difference between a point and a vector?

I understand that a vector has direction and magnitude whereas a point doesn’t.

However, in the course notes that I am using, it is stated that a point is the same as a vector.

Also, can you do cross product and dot product using two points instead of two vectors? I don’t think so, but my roommate insists yes, and I’m kind of confused now.

Here’s an answer without using symbols.

The difference is precisely that between location and displacement.

• Points are locations in space.
• Vectors are displacements in space.

An analogy with time works well.

• Times, (also called instants or datetimes) are locations in time.
• Durations are displacements in time.

So, in time,

• 4:00 p.m., noon, midnight, 12:20, 23:11, etc. are times
• +3 hours, -2.5 hours, +17 seconds, etc., are durations

Notice how durations can be positive or negative; this gives them “direction” in addition to their pure scalar value. Now the best way to mentally distinguish times and durations is by the operations they support

• Given a time, you can add a duration to get a new time (3:00 + 2 hours = 5:00)
• You can subtract two times to get a duration (7:00 – 1:00 = 6 hours)
• You can add two durations (3 hrs, 20 min + 6 hrs, 50 min = 10 hrs, 10 min)

But you cannot add two times (3:15 a.m. + noon = ???)

Let’s carry the analogy over to now talk about space:

• $$(3,5)(3,5)$$, $$(-2.25,7)(-2.25,7)$$, $$(0,-1)(0,-1)$$, etc. are points
• $$\langle 4,-5 \rangle\langle 4,-5 \rangle$$ is a vector, meaning 4 units east then 5 south, assuming north is up (sorry residents of southern hemisphere)

Now we have exactly the same analogous operations in space as we did with time:

• You can add a point and a vector: Starting at $$(4,5)(4,5)$$ and going $$\langle -1,3 \rangle\langle -1,3 \rangle$$ takes you to the point $$(3,8)(3,8)$$
• You can subtract two points to get the displacement between them: $$(10,10) – (3,1) = \langle 7,9 \rangle(10,10) - (3,1) = \langle 7,9 \rangle$$, which is the displacement you would take from the second location to get to the first
• You can add two displacements to get a compound displacement: $$\langle 1,3 \rangle + \langle -5,8 \rangle = \langle -4,11 \rangle\langle 1,3 \rangle + \langle -5,8 \rangle = \langle -4,11 \rangle$$. That is, going 1 step north and 3 east, THEN going 5 south and 8 east is the same thing and just going 4 south and 11 east.

But you cannot add two points.

In more concrete terms: Moscow + $$\langle\text{200 km north, 7000 km west}\rangle\langle\text{200 km north, 7000 km west}\rangle$$ is another location (point) somewhere on earth. But Moscow + Los Angeles makes no sense.

To summarize, a location is where (or when) you are, and a displacement is how to get from one location to another. Displacements have both magnitude (how far to go) and a direction (which in time, a one-dimensional space, is simply positive or negative). In space, locations are points and displacements are vectors. In time, locations are (points in) time, a.k.a. instants and displacements are durations.

EDIT 1: In response to some of the comments, I should point out that 4:00 p.m. is NOT a displacement, but “+4 hours” and “-7 hours” are. Sure you can get to 4:00 p.m. (an instant) by adding the displacement “+16 hours” to the instant midnight. You can also get to 4:00 p.m. by adding the diplacement “-3 hours” to 7:00 p.m. The source of the confusion between locations and displacements is that people mentally work in coordinate systems relative to some origin (whether $$(0,0)(0,0)$$ or “midnight” or similar) and both of these concepts are represented as coordinates. I guess that was the point of the question.

EDIT 2: I added some text to make clear that durations actually have direction; I had written both -2.5 hours and +3 hours earlier, but some might have missed that the negative encapsulated a direction, and felt that a duration is “only a scalar” when in fact the adding of a $$++$$ or $$–-$$ really does give it direction.

EDIT 3: A summary in table form:

+--------------------+------------------------+-----------------------+
| Concept            | SPACE                  | TIME                  |
+--------------------+------------------------+-----------------------+
| LOCATION           | POINT                  | TIME                  |
| DISPLACEMENT       | VECTOR                 | DURATION              |
+--------------------+------------------------+-----------------------+
| Loc - Loc = Disp   | Pt - Pt = Vec          | Time - Time = Dur     |
|                    | (3,5)-(10,2) = <-7,3>  | 7:30 - 1:15 = 6hr15m  |
+--------------------+------------------------+-----------------------+
| Loc + Disp = Loc   | Pt + Vec = Pt          | Time + Dur = Time     |
|                    | (10,2)+<-7,3> = (3,5)  | 3:15 + 2hr = 5:15     |
+--------------------+------------------------+-----------------------+
| Disp + Disp = Disp | Vec + Vec = Vec        | Dur + Dur = Dur       |
|                    | <8,-5>+<-7,3> = <1,-2> | 3hr + 5hr = 8hr       |
+--------------------+------------------------+-----------------------+