Proving trig identities with dot and cross products

[cytochrome]

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 a • b in two ways (Namely a •b = abcos(θ) and a • b = 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

a • b = abcos(θ)

a • b = 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 a •b = abcos(α-β) but I have no idea how to relate this to the trig identities!

 

[SammyS]

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 a • b in two ways (Namely a •b = abcos(θ) and a • b = 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

a • b = abcos(θ)

a • b = 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 a •b = 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?

 

[cytochrome]

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

 

[SammyS]

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 a_x = a*cos(α) and a_y = a*sin(α)

etc.

B.T.W. a_x = a₁, etc. in your Relevant equations for the dot & cross products.

 

[vela]

What are a₁, a₂, b₁, and b₂ in terms of a, b, α, and β?

 

[cytochrome]

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)

 

[SammyS]

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.