# Vector addition

**URL:** <https://boards.straightdope.com/t/vector-addition/88968>\
**Category:** Factual Questions\
**Created:** [October 22, 2001, 11:03pm UTC](https://boards.straightdope.com/t/vector-addition/88968 "2001-10-22T23:03:28Z")\
**Posts on this page:** 20\
**Page:** 1

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**Author:** ![treis](https://avatars.discourse-cdn.com/v4/letter/t/bc79bd/32.png) [@treis](https://boards.straightdope.com/u/treis)\
**Post date:** [October 22, 2001, 11:03pm UTC](https://boards.straightdope.com/t/vector-addition/88968/1 "2001-10-22T23:03:28Z")

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The question is, The angle between the two vectors will be \_\_\_\_\_ when the resultant vector is greatest.

0 or 180

Me says, 180

vector A is ----\>  
vector B is -----\>  
----\>-----\>  
Makes sense they are a straight line making the angle 180 right? WRONG.  
Wha? 0 degrees? how?

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**Author:** ![micco](https://avatars.discourse-cdn.com/v4/letter/m/5f8ce5/32.png) [@micco](https://boards.straightdope.com/u/micco)\
**Post date:** [October 22, 2001, 11:06pm UTC](https://boards.straightdope.com/t/vector-addition/88968/2 "2001-10-22T23:06:52Z")

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Think of the vector direction like a compass heading. If A is aligned due east (90 degrees) and B is 180 degrees from that (270 degrees, due west) then B doubles back on A. If the angle between them is zero, then they both point in the same direction.

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**Author:** ![LordVor](https://avatars.discourse-cdn.com/v4/letter/l/90ced4/32.png) [@LordVor](https://boards.straightdope.com/u/LordVor)\
**Post date:** [October 22, 2001, 11:29pm UTC](https://boards.straightdope.com/t/vector-addition/88968/3 "2001-10-22T23:29:06Z")

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Or, alternately, imagine the vector is your personal velocity. You walk A, which is due east. Then you walk B, which is also due east. How many degrees did you turn to transition from A to B? 0.

-LV

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [October 22, 2001, 11:31pm UTC](https://boards.straightdope.com/t/vector-addition/88968/4 "2001-10-22T23:31:43Z")

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If the angle between A and B is 180 degrees, B = -A and A + B = 0.

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**Author:** ![treis](https://avatars.discourse-cdn.com/v4/letter/t/bc79bd/32.png) [@treis](https://boards.straightdope.com/u/treis)\
**Post date:** [October 22, 2001, 11:37pm UTC](https://boards.straightdope.com/t/vector-addition/88968/5 "2001-10-22T23:37:28Z")

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> [@](#):
>
> Or, alternately, imagine the vector is your personal velocity. You walk A, which is due east. Then you walk B, which is also due east. How many degrees did you turn to transition from A to B? 0.

> [@](#):
>
> Think of the vector direction like a compass heading. If A is aligned due east (90 degrees) and B is 180 degrees from that (270 degrees, due west) then B doubles back on A. If the angle between them is zero, then they both point in the same direction.

Right, but they didn’t ask for the bearing or the direction. They asked for the angle **BETWEEN** the two vectors.

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**Author:** ![panamajack](https://avatars.discourse-cdn.com/v4/letter/p/47e85d/32.png) [@panamajack](https://boards.straightdope.com/u/panamajack)\
**Post date:** [October 22, 2001, 11:39pm UTC](https://boards.straightdope.com/t/vector-addition/88968/6 "2001-10-22T23:39:57Z")

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Note that **ultrafilter** ’s statements only apply to vectors whose magnitudes are equal. ‘A’ and ‘B’ as shown in the OP do not have equal magnitude.

What is generally true is that if the angle between the two is 0, |A + B| = |A| + |B| . If 180, |A + B | = abs ( |A| - |B|).

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**Author:** ![DSeid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dseid/32/20194_2.png) [@DSeid](https://boards.straightdope.com/u/DSeid)\
**Post date:** [October 22, 2001, 11:42pm UTC](https://boards.straightdope.com/t/vector-addition/88968/7 "2001-10-22T23:42:35Z")

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Vectors always have direction. They are not just lines.

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**Author:** ![panamajack](https://avatars.discourse-cdn.com/v4/letter/p/47e85d/32.png) [@panamajack](https://boards.straightdope.com/u/panamajack)\
**Post date:** [October 22, 2001, 11:44pm UTC](https://boards.straightdope.com/t/vector-addition/88968/8 "2001-10-22T23:44:00Z")

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**treis** , the comments were trying to teach you how to figure out the angle between two vectors if you’re having difficulty. So you should preface the posts with, “If you’re trying to figure out the angle BETWEEN two vectors …”

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [October 22, 2001, 11:49pm UTC](https://boards.straightdope.com/t/vector-addition/88968/9 "2001-10-22T23:49:23Z")

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> [@](#):
>
> \*Originally posted by panamajack \*  
> **Note that ultrafilter’s statements only apply to vectors whose magnitudes are equal. ‘A’ and ‘B’ as shown in the OP do not have equal magnitude.**

Yeesh…how’d I miss that?

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**Author:** ![douglips](https://avatars.discourse-cdn.com/v4/letter/d/e68b1a/32.png) [@douglips](https://boards.straightdope.com/u/douglips)\
**Post date:** [October 22, 2001, 11:55pm UTC](https://boards.straightdope.com/t/vector-addition/88968/10 "2001-10-22T23:55:32Z")

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> [@](#):
>
> \*Originally posted by treis \*  
> \*\*
> 
> > [@](#):
> >
> > Or, alternately, imagine the vector is your personal velocity. You walk A, which is due east. Then you walk B, which is also due east. How many degrees did you turn to transition from A to B? 0.
> 
> > [@](#):
> >
> > Think of the vector direction like a compass heading. If A is aligned due east (90 degrees) and B is 180 degrees from that (270 degrees, due west) then B doubles back on A. If the angle between them is zero, then they both point in the same direction.
> 
> Right, but they didn’t ask for the bearing or the direction. They asked for the angle **BETWEEN** the two vectors. \*\*

In order to measure the angle between vectors, put them tail-to-tail. If you do that with your vectors, you’ll see the angle is 0 degrees.  
If you don’t see that the angle is 0 degrees, imagine that B is pointing up just 10 degrees. Put them tail to tail. What’s the angle between them? 10 degrees. As you decrease the ‘upness’ of B, the angle tends to 0.

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<div class="post-metadata">

**Author:** ![treis](https://avatars.discourse-cdn.com/v4/letter/t/bc79bd/32.png) [@treis](https://boards.straightdope.com/u/treis)\
**Post date:** [October 23, 2001, 2:24am UTC](https://boards.straightdope.com/t/vector-addition/88968/11 "2001-10-23T02:24:24Z")

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> [@](#):
>
> originally posted by panamajack  
> treis, the comments were trying to teach you how to figure out the angle between two vectors if you’re having difficulty. So you should preface the posts with, “If you’re trying to figure out the angle BETWEEN two vectors …”

From the O.P.

> [@](#):
>
> The angle **between** the two vectors will be \_\_\_\_\_ when the resultant vector is greatest.

Douglips- I see sort of, why add tail to tail though. They are put together tail to tip. I would measure it from a point on the first vector, to the point at which they meet, and a point on the second vector, why is this wrong?

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<div class="post-metadata">

**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [October 23, 2001, 2:35am UTC](https://boards.straightdope.com/t/vector-addition/88968/12 "2001-10-23T02:35:10Z")

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> [@](#):
>
> \*Originally posted by treis \*  
> \*\*
> 
> > [@](#):
> >
> > originally posted by panamajack  
> > treis, the comments were trying to teach you how to figure out the angle between two vectors if you’re having difficulty. So you should preface the posts with, “If you’re trying to figure out the angle BETWEEN two vectors …”
> 
> From the O.P.
> 
> > [@](#):
> >
> > The angle **between** the two vectors will be \_\_\_\_\_ when the resultant vector is greatest.
> 
> Douglips- I see sort of, why add tail to tail though. They are put together tail to tip. I would measure it from a point on the first vector, to the point at which they meet, and a point on the second vector, why is this wrong? \*\*

Let’s take two vectors that look like the ones in the OP. I’m gonna use (1, 0) and (2, 0). From linear algebra, we have that the cosine of the angle between the two vectors is their dot product divided by the product of their lengths. The dot product is two, and the product of their lengths is 2. So the cosine of the angle between them is 1. Therefore, the angle between them is 2k[sym]p[/sym], and we take k = 0 to use the standard values. Make sense?

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**Author:** ![micco](https://avatars.discourse-cdn.com/v4/letter/m/5f8ce5/32.png) [@micco](https://boards.straightdope.com/u/micco)\
**Post date:** [October 23, 2001, 2:39am UTC](https://boards.straightdope.com/t/vector-addition/88968/13 "2001-10-23T02:39:18Z")

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> [@](#):
>
> \*Originally posted by treis \*  
> **Douglips- I see sort of, why add tail to tail though. They are put together tail to tip. I would measure it from a point on the first vector, to the point at which they meet, and a point on the second vector, why is this wrong?**

Vectors have magnitude and direction. Let’s ignore magnitude and just consider direction. Say that vector A has a direction due east and vector B is due west. That means the vectors have an angle of 180 degrees between them. If you walk out vector A, then turn the 180 degree angle and walk back vector B, you’ve backtracked. So, if you lay vector B’s tail on A’s tip, you end up reversing direction.

```auto

vector A: ----->
vector B: <----- (180 deg difference)

```

If vectors A and B are both due east so they have zero degrees difference between their directions, then you walk out vector A, don’t turn at all since the angle change between A and B is zero, and continue walking east along B. That is, if you lay B’s tail at A’s tip, they keep going in the same direction.

```auto

vector A: ----->
vector B: -----> (zero deg difference)

```

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<div class="post-metadata">

**Author:** ![treis](https://avatars.discourse-cdn.com/v4/letter/t/bc79bd/32.png) [@treis](https://boards.straightdope.com/u/treis)\
**Post date:** [October 23, 2001, 2:46am UTC](https://boards.straightdope.com/t/vector-addition/88968/14 "2001-10-23T02:46:38Z")

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Yeah but micco that is the angle of direction in relation to the first vector. Lets ignore direction for a bit, if I draw a straight line and pick 3 points, say A B and C. Angle ABC will be 180 degrees. Why is it different for vectors?

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [October 23, 2001, 2:57am UTC](https://boards.straightdope.com/t/vector-addition/88968/15 "2001-10-23T02:57:03Z")

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> [@](#):
>
> \*Originally posted by treis \*  
> \*\*Yeah but micco that is the angle of direction in relation to the first vector. Lets ignore direction for a bit, if I draw a straight line and pick 3 points, say A B and C. Angle ABC will be 180 degrees. Why is it different for vectors? \*\*

AB and BC are line segments, and have no direction associated with them (i.e., AB is the same as BA). The vector from A to B is different from the vector from B to A. The formula I gave earlier is what you should use for calculating angles.

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<div class="post-metadata">

**Author:** ![treis](https://avatars.discourse-cdn.com/v4/letter/t/bc79bd/32.png) [@treis](https://boards.straightdope.com/u/treis)\
**Post date:** [October 23, 2001, 3:04am UTC](https://boards.straightdope.com/t/vector-addition/88968/16 "2001-10-23T03:04:41Z")

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Why is it different though ultra I don’t get it. Why can’t I pick a point on one vector, the meeting point, and one on the other vector and calculate the angle that way?

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**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\
**Post date:** [October 23, 2001, 3:43am UTC](https://boards.straightdope.com/t/vector-addition/88968/17 "2001-10-23T03:43:46Z")

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A vector doesn’t have its own position. You can move a vector around anywhere you want, and as long as it has the same magnitude and direction, it’s the same vector. You put them wherever it’s convenient. For adding two vectors together, it’s convenient to put them head to tail. For finding the angle between them, it’s convenient to put them tail to tail.

You could, if you liked, define the angle between two vectors the way you did, but that’s not the way everyone else does it, so you’d confuse people. The main reason why it’s not defined that way is because your definition would mean that any vector is at an angle of 180[sup]o[/sup] from itself, which most folks would consider a bit odd.

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<div class="post-metadata">

**Author:** ![micco](https://avatars.discourse-cdn.com/v4/letter/m/5f8ce5/32.png) [@micco](https://boards.straightdope.com/u/micco)\
**Post date:** [October 23, 2001, 2:01pm UTC](https://boards.straightdope.com/t/vector-addition/88968/18 "2001-10-23T14:01:12Z")

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> [@](#):
>
> \*Originally posted by treis \*  
> **Why is it different though ultra I don’t get it. Why can’t I pick a point on one vector, the meeting point, and one on the other vector and calculate the angle that way?**

You’re treating the vectors like line segments without any direction. Direction is an inherent property in vectors, so if two vectors line up in a straight line, you have to consider which way each of them points in order to determine the angle between them.

```auto

  A B
----->-----> zero degrees

  A B
-----><----- 180 degrees

```

In order to do vector addition, go back to the “walking the line” exercise. If you start at one point and walk off the direction and magnitude of vector A and then turn and walk off the direction and magnitude of vector B, you’re at a point which is the sum of the two vectors in relation to your starting point. So, if you take the first diagram above where the angle between the two is zero, you walk along A, don’t turn at all, and walk along B. If you walked the second example, you’d walk along A, turn 180 degrees and walk along B. The second case is backtracking, so the magnitude of the resulting vector is less than in the first case.

I’m sorry if I keep explaining it the same way. I can’t find better words or examples to make it more clear.

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<div class="post-metadata">

**Author:** ![panamajack](https://avatars.discourse-cdn.com/v4/letter/p/47e85d/32.png) [@panamajack](https://boards.straightdope.com/u/panamajack)\
**Post date:** [October 23, 2001, 5:27pm UTC](https://boards.straightdope.com/t/vector-addition/88968/19 "2001-10-23T17:27:16Z")

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My apologies, **treis**. What I meant was, “preface the **ir** posts”, that is, **micco** ’s & **LordVor** ’s. Because it appears the concept you were missing is exactly what ‘the angle between two vectors’ is. This is what **douglips** was talking about too – not adding them tail to tail (which would be, as you pointed out, incorrect), but to first figure out the angle between them. See also **Chronos** ’s explanation.

I assume the question was posed to you with the two choices you gave. So consider each one in turn. Give your own description of what the vectors would look like, if the angle between them were 0. What would they look like if the angle between them were 180? If you think they ought to be the same, then this is either a much-to-be-hated trick question (and a good argument against trick questions when learning new material) or you might need to go over what you’ve learned.

If this question wasn’t set up in that way, then never mind about the preceding paragraph.

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<div class="post-metadata">

**Author:** ![treis](https://avatars.discourse-cdn.com/v4/letter/t/bc79bd/32.png) [@treis](https://boards.straightdope.com/u/treis)\
**Post date:** [October 23, 2001, 8:03pm UTC](https://boards.straightdope.com/t/vector-addition/88968/20 "2001-10-23T20:03:36Z")

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> [@](#):
>
> You could, if you liked, define the angle between two vectors the way you did, but that’s not the way everyone else does it, so you’d confuse people

Thanks Chronos and everyone for thier help it makes sense now. It could be done my way but everyone else does it the other way and therefore it is the “right” way.

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