# Proving trig identities with dot and cross products

## 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)

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!

SammyS
Staff Emeritus
Homework Helper
Gold Member

## 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)

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?

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
Staff Emeritus
Homework Helper
Gold Member
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.

vela
Staff Emeritus
Homework Helper
What are a1, a2, b1, and b2 in terms of a, b, α, and β?

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
Staff Emeritus
Homework Helper
Gold Member
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.