r1 and r2 are unit vectors in the x-y plane making angles a and b with the positive x-axis.by considering r1.r2 derive
cos(a-b)=cos(a)cos(b)+sin(a)sin(b)
To derive cos(a - b) using the dot product of unit vectors r1 and r2, we can start by representing r1 and r2 in terms of their respective components.
Let's assume r1 = (r1x, r1y) and r2 = (r2x, r2y).
Given that r1 and r2 are unit vectors, their magnitudes are equal to 1. Therefore, we have:
|r1| = 1 and |r2| = 1.
Now, let's consider the dot product of r1 and r2, denoted as r1 · r2.
r1 · r2 = r1x * r2x + r1y * r2y.
We can rewrite r1 and r2 in terms of their angles as follows:
r1x = cos(a), r1y = sin(a),
r2x = cos(b), r2y = sin(b).
Substituting these values into the equation for the dot product:
r1 · r2 = (cos(a) * cos(b)) + (sin(a) * sin(b)).
Recall that cos(a - b) = cos(a) * cos(b) + sin(a) * sin(b).
Comparing this to the equation we obtained above, we can see that:
cos(a - b) = r1 · r2.
Hence, by considering the dot product of r1 and r2, we have derived the equation cos(a - b) = cos(a) * cos(b) + sin(a) * sin(b).
To derive cos(a-b) using the given information about r1 and r2, we can use the dot product formula:
r1 · r2 = |r1| |r2| cos(θ),
where r1 · r2 is the dot product of r1 and r2, |r1| and |r2| are the magnitudes of r1 and r2, and θ is the angle between r1 and r2.
Since r1 and r2 are unit vectors, their magnitudes are both equal to 1. Therefore, the above formula simplifies to:
r1 · r2 = cos(θ).
Now, let's determine the angles between r1, r2, and the positive x-axis:
- Angle a is the angle between r1 and the positive x-axis.
- Angle b is the angle between r2 and the positive x-axis.
Since cos(a) is the cosine of the angle between r1 and the positive x-axis, we can rewrite this as:
cos(a) = cos(angle between r1 and the positive x-axis) = cos(angle a).
Similarly, we can rewrite cos(b) as:
cos(b) = cos(angle b).
Now, let's find the angle between r1 and r2, denoted as θ:
θ = angle between r1 and r2 = a - b.
Using the dot product formula, we can write:
r1 · r2 = cos(θ),
which can be rewritten as:
cos(a - b) = r1 · r2.
Since r1 and r2 are unit vectors, their magnitudes are both equal to 1. Therefore, we can substitute r1 · r2 with:
cos(a - b) = cos(a) cos(b) + sin(a) sin(b).
Hence, we have derived:
cos(a - b) = cos(a) cos(b) + sin(a) sin(b).