Angle Between 2 Vectors

1. Feb 16, 2009

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 im 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 dont get why the cross product would give a smaller angle? Can anyone tell me

If i take 114 - 180 i get -66 but I dont get why I would subtract 180 *and also its a negative angle then..HELP!

Last edited: Feb 16, 2009
2. Feb 16, 2009

malawi_glenn

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

huh?

3. Feb 16, 2009

salman213

4. Feb 16, 2009

malawi_glenn

Can you show us how you did?

5. Feb 16, 2009

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

Last edited: Feb 16, 2009
6. Feb 16, 2009

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)

7. Feb 16, 2009

salman213

im sorry i drew it out but i dont 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 [Broken]

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...:(

Last edited by a moderator: May 4, 2017
8. Feb 16, 2009

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?

9. Feb 16, 2009

salman213

yea sorry i put int he wrong component..its

S =ax + 2ay + az,

10. Feb 16, 2009

salman213

^now the new picture looks like the angle IS 114 but i dont see how the smallestis 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!

11. Feb 16, 2009

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!

12. Feb 16, 2009

dontdisturbmycircles

And you're drawing is incorrect.

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

13. Feb 16, 2009

salman213

but cos of (66 ) = 0.4

cos of (114 ) = -0.4

they are not the same

14. Feb 16, 2009

dontdisturbmycircles

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

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

15. Feb 16, 2009

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 $$\theta > 90$$°

16. Feb 16, 2009

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

both?

17. Feb 16, 2009

malawi_glenn

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

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

18. Feb 16, 2009

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.

19. Feb 16, 2009

salman213

yes thats what i was wondering isnt 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.

20. Feb 16, 2009

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. Lets 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, thats why you should use the dot product.