Vector A has a magnitude of 7 and is at an angle of 22.5 degrees measured counter-clockwise from the x axis. Vector B has a magnitude of 12 and is at an angle of 110.4 degrees measured counter-clockwise from the x axis. Vector C has a magnitude of 6 and is at an angle of 180 degrees measured counter-clockwise from the x axis. Lastly, Vector D has a magnitude of 17 and is at an angle of 280.1 degrees measured counter-clockwise from the x axis. If R = A - B - C + D, find the magnitude of R.

To find the magnitude of vector R, we need to first find the components of each vector A, B, C, and D, and then add them up.

Let's start by finding the x and y components of vector A:

The x-component of A = magnitude of A * cos(angle of A)
= 7 * cos(22.5 degrees)

The y-component of A = magnitude of A * sin(angle of A)
= 7 * sin(22.5 degrees)

Note that we are using the trigonometric functions cosine (cos) and sine (sin) since the angle is measured counter-clockwise from the x-axis.

Similarly, we can find the x and y components of vectors B, C, and D using the same equations.

For vector B:
x-component of B = magnitude of B * cos(angle of B)
= 12 * cos(110.4 degrees)
y-component of B = magnitude of B * sin(angle of B)
= 12 * sin(110.4 degrees)

For vector C:
x-component of C = magnitude of C * cos(angle of C)
= 6 * cos(180 degrees)
y-component of C = magnitude of C * sin(angle of C)
= 6 * sin(180 degrees)

For vector D:
x-component of D = magnitude of D * cos(angle of D)
= 17 * cos(280.1 degrees)
y-component of D = magnitude of D * sin(angle of D)
= 17 * sin(280.1 degrees)

Now, we can add up the x and y components separately to find the components of vector R:
x-component of R = x-component of A - x-component of B - x-component of C + x-component of D
y-component of R = y-component of A - y-component of B - y-component of C + y-component of D

Finally, we can find the magnitude of vector R using the components:
magnitude of R = sqrt((x-component of R)^2 + (y-component of R)^2)

By plugging in the values and performing the calculations, we can find the magnitude of vector R.