A Force's Direction Angle Calculation

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.
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