What is the direction angle between force Q of magnitude 450N directed from C(-3,4,0) to D(1,5,3) and the z-coordinate axis?
The direction angle between the force Q and the z-coordinate axis is approximately 53.96 degrees. The correct answer is option b.
Detailed Explanation:
Given data:
Force Q magnitude: 450N
Points C(-3, 4, 0) and D(1, 5, 3)
Calculations:
The components of force Q can be determined by finding 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 force Q: Qx = 4 N, Qy = 1 N, and Qz = 3 N.
To find the direction angle with respect to the z-axis, we use the trigonometric relationship:
cos(θ_z) = Qz / |Q|
The magnitude of force Q, denoted by |Q|, can be calculated as:
|Q| = sqrt(Qx² + Qy² + Qz²)
|Q| = sqrt(4² + 1² + 3²) = sqrt(16 + 1 + 9) = sqrt(26)
Next, calculate the cosine of the direction angle:
cos(θ_z) = 3 / sqrt(26)
Finally, determine the direction angle with respect to the z-axis:
θ_z = cos^(-1)(cos(θ_z))
θ_z ≈ 53.96 degrees
Therefore, the correct direction angle between force Q and the z-coordinate axis is approximately 53.96 degrees (Option B). For further understanding of angles, you can refer to external sources.