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Proving trig identities with dot and cross products

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Homework Statement



The two vectors a and b lie in the xy plane and make angles α and β with the x axis.

a)By evaluating ab in two ways (Namely ab = abcos(θ) and ab = a1b1+a2b2) prove the well-known trig identity
cos(α-β)=cos(α)cos(β)+sin(α)sin(β)

b)By similarly evaluating a X b prove that
sin(α-β) = sin(α)cos(β) - cos(α)sin(β)

c)Now let vector a make an angle -α with the x axis and find a similar expression for
cos(α+β)


Homework Equations



ab = abcos(θ)

ab = a1b1+a2b2



The Attempt at a Solution



I drew the vectors a and b with their appropriate angles to the x-axis... The angle between the vectors is (α-β) so I have ab = abcos(α-β) but I have no idea how to relate this to the trig identities!
 

Answers and Replies

  • #2
SammyS
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Homework Statement



The two vectors a and b lie in the xy plane and make angles α and β with the x axis.

a)By evaluating ab in two ways (Namely ab = abcos(θ) and ab = a1b1+a2b2) prove the well-known trig identity
cos(α-β)=cos(α)cos(β)+sin(α)sin(β)

b)By similarly evaluating a X b prove that
sin(α-β) = sin(α)cos(β) - cos(α)sin(β)

c)Now let vector a make an angle -α with the x axis and find a similar expression for
cos(α+β)

Homework Equations



ab = abcos(θ)

ab = a1b1+a2b2

The Attempt at a Solution



I drew the vectors a and b with their appropriate angles to the x-axis... The angle between the vectors is (α-β) so I have ab = abcos(α-β) but I have no idea how to relate this to the trig identities!
What are the components of vector, a, in terms of a (the magnitude of the vector) and the angle α?

What are the components of vector, b, in terms of b (the magnitude of the vector) and the angle β?
 
  • #3
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What are the components of vector, a, in terms of a (the magnitude of the vector) and the angle α?

What are the components of vector, b, in terms of b (the magnitude of the vector) and the angle β?
The components of a are ax*cos(α) and ay*sin(α)

The components of b are bx*cos(β) and by*sin(β)

Thanks for reminding me about that I'll see where I can get from here
 
  • #4
SammyS
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The components of a are ax*cos(α) and ay*sin(α)

The components of b are bx*cos(β) and by*sin(β)

Thanks for reminding me about that I'll see where I can get from here
Not quite right.

The components of a are ax = a*cos(α) and ay = a*sin(α)

etc.

B.T.W. ax = a1, etc. in your Relevant equations for the dot & cross products.
 
  • #5
vela
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What are a1, a2, b1, and b2 in terms of a, b, α, and β?
 
  • #6
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What are a1, a2, b1, and b2 in terms of a, b, α, and β?
a1 = acos(α)
a2 = asin(α)

b1 = bcos(β)
b2 = bsin(β)

a dot b = acos(α)bcos(β) + asin(α)bsin(β)

which pretty much completes the proof for a)
 
  • #7
SammyS
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a1 = acos(α)
a2 = asin(α)

b1 = bcos(β)
b2 = bsin(β)

a dot b = acos(α)bcos(β) + asin(α)bsin(β)

which pretty much completes the proof for a)
Do something similar for (b).

( a × b )3 = a1*b2 - a2*b1 . If both vectors are in the xy-plane, then the other components are zero.
 

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