How Do You Calculate the Angle Between Two Vectors in a Sailboat Rigging System?

[rafael_josem]

Homework Statement

The figure shows a mast and parts of the equipments of a sailboat. The elements CD and EF belongs to the same plane, CD has 7.5m of length and has an angle of 45º with a vertical line that passes by C. When $$\theta$$ = 15º the tension in the rope AB is 230N.

http://img128.imageshack.us/img128/8149/problemas3.th.jpg

a) find the angle between the ropes AB and BD

Homework Equations

ABcos($$\theta$$) = A.B

Where bold letters represent vectors.

The Attempt at a Solution

I started by finding the coordinates of points A, B and D.

A = (0.5,0,-3)
B = (x, 22, 0)
D = ?

I get lost finding the x coordinate for B and the coordinate of D. With these coordinates I can find the vectors that I need to apply the equation above by doing the following:

AB = (x - 0.5)i + (22-0)j + (0 + 3)k

Any ideas?

Thanks

 

[anathema]

for your post! To find the coordinates of D, you can use the given information about the length and angle of CD. Since CD has a length of 7.5m and an angle of 45º with a vertical line, you can use trigonometry to find the coordinates of D. Remember that the x coordinate of D will be the same as the x coordinate of C, since they both lie on the same vertical line.

Once you have the coordinates of D, you can use the same method to find the coordinates of B. Since you know the length of AB and the angle \theta, you can use trigonometry to find the x coordinate of B.

Once you have all the coordinates, you can find the vectors AB and BD and use the equation ABcos(\theta) = A.B to find the angle between the two ropes. I hope this helps!

 

[m2003]

for your question. Finding the angle between vectors can be helpful in many situations, including this one with the sailboat. In order to solve this problem, it would be helpful to use vector notation to represent the given information. We can express the vectors AB, BD, and CD as follows:

AB = (x - 0.5)i + (22-0)j + (0 + 3)k
BD = (x - ?)i + (22- ?)j + (0 + ?)k
CD = (x - ?)i + (22- ?)j + (0 + ?)k

Using the given information, we can also write the following equations:

|AB|cos(15) = 230
|CD|cos(45) = 7.5

Now, we can solve for the unknown coordinates of B and D by setting up a system of equations using the above equations and the fact that the angle between AB and BD is the same as the angle between CD and EF. Once we have the coordinates of B and D, we can use the dot product formula to find the angle between AB and BD. I hope this helps!