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Answer :
Final answer:
The approximate components calculated are x = 220 N, y = 110 N, z = 240 N. To determine the components of a force vector with a given magnitude and direction angles, trigonometric functions are used (Option E).
Explanation:
To determine the x, y, and z components of a force vector f given its magnitude and angles in a three-dimensional space, we use trigonometric functions.
Given f = 340 N, β = 65° (angle from the y-axis), and γ = 50° (angle from the z-axis), we can find the components using the following formulas:
- fx = f * cos(γ)
- fy = f * sin(γ) * cos(β)
- fz = f * sin(γ) * sin(β)
Plugging in the values:
- fx = 340 N * cos(50°) = 340 N * 0.6428 = 218.5 N (approximately 220 N)
- fy = 340 N * sin(50°) * cos(65°) = 340 N * 0.7660 * 0.4226 = 110.2 N (approximately 110 N)
- fz = 340 N * sin(50°) * sin(65°) = 340 N * 0.7660 * 0.9063 = 236.5 N (approximately 240 N)
Following the trigonometric calculations accurately with the given angles leads to the components 220 N (x), 110 N (y), and 240 N (z).
Hence, the correct answer is: (Option E).
The complete question is:
If f = 340 N, β = 65°, and γ = 50°, determine the x, y, z components of f.
A) x = 170 N, y = 220 N, z = 290 N
B) x = 220 N, y = 170 N, z = 290 N
C) x = 290 N, y = 220 N, z = 170 N
D) x = 170 N, y = 290 N, z = 220 N
E) x = 220 N, y = 110 N, z = 240 N
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