Question Use the image to answer the question. Vectors u and v are graphed. Explain in detail each step necessary to find the angle between the two vectors and find the angle in degrees. (4 points)

Answer 1

To find the angle between two vectors u and v, you can use the formula:

cos(theta) = (u dot v) / (||u|| * ||v||)

Where "u dot v" represents the dot product of the two vectors, and "||u||" and "||v||" represent the magnitudes of the vectors.

1. First, calculate the dot product of the two vectors:
u dot v = (u1 * v1) + (u2 * v2)
In this case, assuming the components of vector u are (u1, u2) and the components of vector v are (v1, v2), you can calculate:
u dot v = (2 * 3) + (3 * 4) = 6 + 12 = 18

2. Next, calculate the magnitudes of the two vectors:
||u|| = sqrt(u1^2 + u2^2)
||v|| = sqrt(v1^2 + v2^2)
In this case:
||u|| = sqrt(2^2 + 3^2) = sqrt(4 + 9) = sqrt(13)
||v|| = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5

3. Substitute the values into the formula:
cos(theta) = 18 / (sqrt(13) * 5)

4. Finally, solve for theta:
theta = arccos(18 / (sqrt(13) * 5))

Using a calculator to find the arccosine of the value, you can determine the angle between the two vectors in degrees.