Direction Angle Calculation between Force Vector and Z-Coordinate Axis

What is the direction angle between the force vector and the z-coordinate axis?

A force Q of magnitude 450N is directed from C(-3,4,0) to D(1,5,3). Determine the direction angle between Q and the z-coordinate axis.

Choose the correct option:

A) 35.69

B) 87.96

C) 56.93

D) 78.69

E) 38.33

F) 33.83

G) 83.33

H) 53.96

Correct Answer:

The correct option of the given statement "A force Q of magnitude 450N is directed from C(-3,4,0) to D(1,5,3). Determine the direction angle between Q and the z-coordinate axis." is C) 56.93.

To determine the direction angle between force Q and the z-coordinate axis, we can use the dot product between the force vector and the z-coordinate axis vector. First, let's find the vector representing the force Q. To do this, we subtract the coordinates of point C from the coordinates of point D:

Q = D - C = (1, 5, 3) - (-3, 4, 0) = (4, 1, 3)

Next, we need to find the unit vector in the direction of the z-coordinate axis. The z-coordinate axis vector is (0, 0, 1), but to find the unit vector, we need to divide it by its magnitude, which is 1:

u = (0, 0, 1) / 1 = (0, 0, 1)

Now, we can calculate the dot product between Q and u:

Q · u = (4, 1, 3) · (0, 0, 1) = 0 + 0 + 3 = 3

The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them. In this case, the dot product is equal to 3.

To find the angle between Q and the z-coordinate axis, we can use the inverse cosine function:

cos θ = Q · u / (|Q| |u|)

Substituting the values we have:

cos θ = 3 / (√26 × 1)

θ ≈ arccos(3 / √26) ≈ 56.93 degrees

Therefore, the direction angle between force Q and the z-coordinate axis is approximately 56.93 degrees.