There are two vectors, a and b, making angles alpha and beta respectively with their resultant vector. We need to find the magnitude of the resultant vector.

- Vector Addition:
When two vectors are added, their resultant vector can be found using vector addition. The resultant vector is the vector sum of the two given vectors.

- Law of Cosines:
In this case, we can use the Law of Cosines to find the magnitude of the resultant vector. The Law of Cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the two sides and the cosine of the included angle.

- Calculation:
Let's assume the magnitude of vector a is A, the magnitude of vector b is B, and the magnitude of the resultant vector is R.
According to the Law of Cosines:
R^2 = A^2 + B^2 - 2ABcos(180 - (alpha + beta))
R^2 = A^2 + B^2 + 2ABcos(alpha + beta)

- Angle between Resultant and Vectors:
The angle between the resultant vector and vector a can be found using the Law of Sines or the Law of Cosines. Similarly, the angle between the resultant vector and vector b can also be calculated.

- Final Step:
Once you have the magnitude of the resultant vector, you can also find the direction of the resultant vector by calculating the angle it makes with the x-axis or any other reference axis.

By following these steps and calculations, you can determine the magnitude of the resultant vector when given the angles alpha and beta that vectors a and b make with the resultant vector.