John Taylor Classical Mechanics Chapter 1 Problem 22

[karmonkey98k]

Homework Statement

The two vectors a and b lie in the xy plane and make angles (alpha and beta) with the x axis. a. by evaluating a dot b in two ways prove the well known trig identity cos(alpha-beta)=cosalphacosbeta +sinalphasinbeta

Homework Equations

adotb=abcostheta=axbx+ayby

The Attempt at a Solution

Could not really come to the proper set-up.

 

[ehild]

Homework Statement

The two vectors a and b lie in the xy plane and make angles (alpha and beta) with the x axis. a. by evaluating a dot b in two ways prove the well known trig identity cos(alpha-beta)=cosalphacosbeta +sinalphasinbeta

Homework Equations

adotb=abcostheta=axbx+ayby

The Attempt at a Solution

Could not really come to the proper set-up.

How are the x,y components of a vector related to the angle they make with the x axis?

ehild

 

[karmonkey98k]

the angle they make is in the problem; the cos of the difference of the angles equals that result. i just cud not go from the dot product to that.

 

[tiny-tim]

hi karmonkey98k!

start with a = |a|cosαi + |a|sinαj

 

[Nickie]

I would suggest the following approach to solve this problem:

1. Start by drawing a diagram to visualize the situation. Label the two vectors a and b, and the angles alpha and beta.

2. Write out the dot product formula for a dot b. This will be helpful in understanding the relationship between the two vectors.

3. Use the trigonometric identities for cos(alpha) and sin(alpha) to express the vector a in terms of its x and y components. Similarly, express the vector b in terms of its x and y components.

4. Substitute these expressions into the dot product formula and simplify.

5. Use the trigonometric identities for cos(beta) and sin(beta) to simplify the resulting expression.

6. Compare the final expression with the trig identity cos(alpha-beta)=cosalphacosbeta +sinalphasinbeta. You should see that they are equivalent, proving the identity.

7. To further solidify your understanding, try plugging in different values for alpha and beta and see if the identity holds true.

By following these steps, you should be able to solve this problem and prove the given trig identity.