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:

To determine the direction angle between the force Q and the z-coordinate axis, we need to find the angle between the force vector and the z-axis. First, we find the direction vector of force Q by subtracting the coordinates of point C from point D:

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.
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