What is the correct method for finding the smallest angle between two vectors?

[salman213]

1. Find the SMALLEST angle between the vectors T and S

Given vectors T = 2ax — 6ay + 3az and S =ax + 2ay + az,




See the thing I am confused about is whether to use Cross Product or Dot Product. I used the dot product formula

TdotS = |T||S|cos

and solved for cos theta ((theta = cos-1))

I got 114 degrees

The solution I have uses CROSS PRODUCT and finds an angle 65 Degrees

I don't get why the cross product would give a smaller angle? Can anyone tell me
If i take 114 - 180 i get -66 but I don't get why I would subtract 180 *and also its a negative angle then..HELP!

 

[malawi_glenn]

Given vectors T = 2ax — 6ay + 3az and 8 = 3^-4- 2ay + az,

huh?

 

[salman213]

sorry about that, fixed now..

 

[malawi_glenn]

Can you show us how you did?

 

[salman213]

ok sure its a simple dot product tahts why i didnt show it my main question is can we go from an angle of 114 to 66..and if its because of 180-114 what would be the reason for subtracting 180?Anyways ill show it

Given vectors T = 2ax — 6ay + 3az and S =ax - 2ay + az,

T dot S = 2 -12 + 3 = -7

|T| = sqrt 49 = 7
|S| = sqrt 6

cos -1(-7 / (7 * sqrt(6) ) = 114

the solution I have uses the cross product and the angle they get is 66 degrees

 

[malawi_glenn]

if you draw the situation on a paper, you will loose the confusion.

T dot S = 2 + 12 + 3 = 15 (IT IS 17)

 

[salman213]

im sorry i drew it out but i don't see how this works...

when i take the dot product and cos inverse i get 114 so this is not the angle between the two vectors?

http://img18.imageshack.us/img18/2420/95148869qw3.jpg obviously from the picture it seems the angle is infact 66 degrees but why then mathematically i get 114 degrees using the dot product?

I thought that angle that i get from the dot product is the angle between the vectors so why did I get 114...:(

 

[malawi_glenn]

if the angle is 114 also 180 -114 = 66 degree is the angle between the vectors.

Now I get with my calculator that \text{arccos} (17/(7\sqrt{6})) = 7.5 degree.

Can you outline the solution using cross product for me?

 

[salman213]

yea sorry i put int he wrong component..itsS =ax + 2ay + az,

 

[salman213]

^now the new picture looks like the angle IS 114 but i don't see how the `smallest`is 66 degrees...I do see that the angle 66 degrees is made with the vector S and NEGATIVE T

but S and T is 114...the cross product method is just the general cross product and then take sin -1

I just use my CASIO calculator it does the cross product

THOUGH YOU TAKE MAGNTIUDE OF (T X S) = |T||S| sin angle

sin -1 will give u 66 degrees!

 

[dontdisturbmycircles]

Remember, 114 has the same sin as 66 degrees! Cos is maybe easier to use for this problem as it is uniquely valued in the 0°-180° region. Whenever you use trig formulas, make a habit of remember that ALL the trig formulas are multivalued!

 

[dontdisturbmycircles]

And you're drawing is incorrect.

edit: The 3d plot shown above IS correct, sorry.

 

[salman213]

but cos of (66 ) = 0.4

cos of (114 ) = -0.4 they are not the same

 

[dontdisturbmycircles]

That is correct, they are not, that is why the dot product gives you the correct answer.

cos(\Theta)=cos(-\Theta)

 

[dontdisturbmycircles]

Remember that cosine takes values of 1 to -1 in the 0° to 180° range, but sin is double valued in that region, that is: it takes on each value between 0 and 1 twice. So if you want to use sine, you have to ask yourself at the end of the problem if it could possibly be 114° instead of 66°. If you use cosine, you know you have the right answer. You could check by taking the dot product of the two vectors, you will find that it is negative and hence \theta > 90°

 

[salman213]

im confused still you said the dot product is the correct answer? So on a test if it said find the angle between two vectors..

from the cross product the angle is found to be 66

from the dot product the angle is found to be 114

What is the right answer?

both?

 

[malawi_glenn]

two vectors have two angles, which sum is 180 degrees.

Draw two lines (2D vectors) in the plane.

 

[dontdisturbmycircles]

I have always been asked to find the angle between the two vectors if they are placed tail to tail, which the dot product gives the correct answer for and the cross product does too, but your CALCULATOR gives you the wrong answer for the sin. Use the dot product, this is a classical application of the dot product.

 

[salman213]

yes that``s what i was wondering isn``t the angle used in the dot and cross product the angle between the vectors placed TAIL TO TAIL...

if that is true then the angle is 114.

 

[dontdisturbmycircles]

yes! That's exactly right, the vectors need to be placed tail to tail in all questions I have ever done.

Make sure you understand why the sine didn't give you the correct answer, its only because each angle in the 0° - 180° range is duplicated. Let's say I tell you that the sine of the angle between two vectors is 0.9. Well then you should punch it into your calculator and see that the 'angle' is 64.2° using inverse sine, but I could say it could also be 115.8, it has the same sine! Cos on the other hand only has one answer for these problems, that's why you should use the dot product.

 

[salman213]

Yes thank you i understand that part from the graph like you said, unfortunately the solutions did it using cross product and said the smallest angle is 66. I guess they are wrong or the question was not clear enough to say not necessarily the tail to tail angle.

Thanks again!