The angle θ between the direction of vector A and the positive direction of x can be found using the tangent of the angle θ, which is the ratio of the y-component to the x-component of the vector.
The formula to find the angle is:
θ = atan(|Ay/Ax|)
where Ay and Ax are the y and x components of vector A, respectively.
Substituting the given values:
θ = atan(|40.0/-25.0|)
This gives us θ = atan(1.6).
However, because the x-component is negative and the y-component is positive, this means that the vector is in the second quadrant. The arctangent function typically returns values in the range (-π/2, π/2), which corresponds to the first and fourth quadrants, so we need to add π/2 (or 90 degrees) to get the angle in the correct quadrant.
So, θ = atan(1.6) + π/2.
Using a calculator to find the arctangent and converting from radians to degrees (since 1 radian is approximately 57.3 degrees), we get:
θ ≈ 58 degrees + 90 degrees = **148 degrees**.
So, the angle between the direction of vector A and the positive direction of x is **148 degrees**.