Determining Direction Angle between Force and Z-Coordinate Axis
What is the direction angle between force Q of 450N from C(-3,4,0) to D(1,5,3) and the z-coordinate axis?
G. A force Q of magnitude 450N is directed from C(-3,4,0) to D(1,5,3), the direction angle between Q and the z-coordinate axis is B. 56.93.
Calculating Direction Angle:
Direction vector: (1-(-3), 5-4, 3-0) = (4, 1, 3)
Next, we find the dot product of the direction vector and the unit vector along the z-axis, which is (0, 0, 1):Dot product: (4*0) + (1*0) + (3*1) = 3
The magnitude of the direction vector is given by:Magnitude: √(4^2 + 1^2 + 3^2) = √(16 + 1 + 9) = √26
The angle between two vectors can be found using the formula:Cosine of the angle: (Dot product) / (Magnitude of the direction vector * Magnitude of the z-axis vector)
Plugging in the values:Cosine of the angle = 3 / (√26 * 1) = 3 / √26
Finally, we find the angle by taking the inverse cosine of the value obtained:Angle: arccos(3 / √26) ≈ 56.93 degrees
Therefore, the direction angle between Q and the z-coordinate axis is approximately 56.93 degrees.