What is the direction angle between the force Q and the x-coordinate axis in degrees?

Question

Joey · Answer

Calculation of Direction Angle: The direction angle between the force Q and the x-coordinate axis is approximately 38.33 degrees. The correct answer is option F.  To find the direction angle between the force Q and the x-coordinate axis, we first need to find the components of the force Q in the x, y, and z directions. Given points C(-3, 4, 0) and D(1, 5, 3), the components of force Q can be calculated as the differences in the x, y, and z coordinates between points C and D:  Qx = Dx - Cx = 1 - (-3) = 4 N Qy = Dy - Cy = 5 - 4 = 1 N Qz = Dz - Cz = 3 - 0 = 3 N  Now, we have the components of the force Q: Qx = 4 N, Qy = 1 N, and Qz = 3 N. The direction angles with respect to the x, y, and z axes can be found using trigonometric relationships: cos(θ_x) = Qx / |Q|, cos(θ_y) = Qy / |Q|, cos(θ_z) = Qz / |Q|.   The magnitude of force Q, |Q|, is given by |Q| = sqrt(Qx^2 + Qy^2 + Qz^2). Substitute the values and calculate |Q| = sqrt(26).  Now, find the cosines of the direction angles: cos(θ_x) = 4 / sqrt(26), cos(θ_y) = 1 / sqrt(26), cos(θ_z) = 3 / sqrt(26). Finally, calculate the direction angle with respect to the x-axis: θ_x = cos^(-1)(cos(θ_x)), which is approximately 38.33 degrees.   Therefore, the correct answer for the direction angle between force Q and the x-coordinate axis is approximately 38.33 degrees.

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