Find the sum of the vectors u and v if theta is the angle between them

In summary, the book provides an incorrect answer for the sum of vectors when the angle between them is greater than 90 degrees. To get the correct answer, one must use one of two methods - a scaled drawing or working with component vectors.
  • #1
DevilTemptations
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:eek: After spending about an hour on this problem, I've become completely convinced that my math book is wrong in the answer it provides. Am I wrong or right? This is from the Harcourt Geometry and Discrete Mathematics Text.

Question:
Find the sum of the vectors u and v if theta is the angle between them.

Given:
u=3 (magnitude)
v=10 (magnitude)
theta=115 degrees

Obviously, the angles in a parallelogram equal to 360. This means the other vertexs are 65 degrees. Then I would use the cosine law to solve for the sum of the vectors. (I wish I could draw a diagram but I can't). So...

u+v^2=(10)^2+(3)^2-2(10)(3)Cos 65 degrees
therefore, u+v=9.1 (approx)

However, the back of the book says u+v=11.6

I've found that to get 11.6, instead of putting Cos 65, I'd have to put Cos 115 but that would not follow according to the laws. Another way is instead of finding the sum, I'd actually be finding the DIFFERENCE between the vectors.

Then I thought maybe I'm doing something wrong... because in the previous question, the angle between the vectors was 70 degrees. The angle between the vectors was not REALLY 70, instead it was divided into two angles and 70 degrees was one of them. Thus, an exterior angle from 70 which is 110 is given... so this proves the vertex opposite from it would be 70. Use Cosine law, I got the right answer.

However, this is an obtuse angle. There is obviously a different method but maybe I don't know it.

I'm confused :yuck:

Any ideas within the next hour or two would be appreciated...
 
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  • #2
Well, one method is to make a scaled drawing and determine the resultant graphically. The other is to work with components.Consider the u vector along the x-axis. You would then resolve v vector into it's x-and y components. Add up the x and y components to get the components of the resultant. It's magnitude can determined by applying pythagoras' theorem to the components. The direction of the resultant is determined with the tangent of it's (the resultant's) components.
 
  • #3
If the angle between the vetors is greater than 90 degrees, then the length of their sum will be smaller than the length of either one. Looks to me like the book's answer is the difference of the two vectors.
 
  • #4

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1. What are vectors u and v?

Vectors u and v are mathematical quantities that have both magnitude and direction. They are commonly used in scientific fields such as physics and mathematics to represent physical quantities like velocity and force.

2. How is the sum of vectors u and v calculated?

The sum of vectors u and v is calculated by adding the corresponding components of each vector. This can be done graphically by placing the vectors tip-to-tail and drawing a resultant vector from the tail of the first vector to the tip of the last vector. The magnitude and direction of the resultant vector can also be calculated using trigonometric functions if the angle between the two vectors is known.

3. What is the significance of theta in finding the sum of vectors u and v?

The angle theta represents the direction and relative orientation of the two vectors. It is necessary to know this angle in order to accurately calculate the magnitude and direction of the resultant vector. Without theta, the sum of the vectors cannot be fully determined.

4. Can the sum of vectors u and v be negative?

Yes, the sum of vectors u and v can be negative. This occurs when the two vectors are in opposite directions and cancel each other out, resulting in a negative magnitude for the resultant vector.

5. How is the sum of vectors u and v represented mathematically?

The sum of vectors u and v is represented as u + v. This notation is used in mathematical equations to indicate the addition of the two vectors. Alternatively, the sum can also be represented using vector notation, such as u = (u1, u2) and v = (v1, v2), where u1 and v1 are the x-components of the vectors and u2 and v2 are the y-components.

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