find the angle theta between two vectors v=-sq. root of 3/2 i+-square root of 3 / 2 j and w=4 square root of 3 i +4j

To find the angle theta between two vectors, you can use the dot product formula and the magnitude of each vector. Here are the steps to calculate the angle between vectors v and w:

Step 1: Calculate the dot product of the two vectors.
- The dot product (⋅) of two vectors v and w is given by: v ⋅ w = |v| * |w| * cos(theta).
- In our case, v = (-√3/2)i + (√3/2)j and w = (4√3)i + 4j.
- Calculate the magnitude of vector v: |v| = √((-√3/2)^2 + (√3/2)^2).
- Calculate the magnitude of vector w: |w| = √((4√3)^2 + 4^2).
- Calculate the dot product: v ⋅ w = (-√3/2 * 4√3) + (√3/2 * 4).

Step 2: Calculate the magnitudes of each vector.
- Calculate the magnitude of vector v: |v| = √((-√3/2)^2 + (√3/2)^2).
- Calculate the magnitude of vector w: |w| = √((4√3)^2 + 4^2).

Step 3: Calculate the angle theta.
- Use the formula: cos(theta) = (v ⋅ w) / (|v| * |w|).
- Substitute the values we calculated: cos(theta) = ((-√3/2 * 4√3) + (√3/2 * 4)) / (√((-√3/2)^2 + (√3/2)^2) * √((4√3)^2 + 4^2)).
- Solve for theta by taking the inverse cosine (arccos) of cos(theta): theta = arccos(cos(theta)).

Now, plug in the values into a calculator to find the exact value of theta.