A force of 500 N forms angles of 60°, 45°, and 120°, respectively, with the x, y, and z axes. Find the components [latex]F_x, F_y, \text { and } F_z[/latex] of the force and express the force in terms of unit vectors.

Substitute [latex]F=500 \mathrm{~N}, \theta_x=60^{\circ}, \theta_y=45^{\circ}, \text { and } \theta_z=120^{\circ}[/latex] formulas (2.19). The scalar components of F are then [latex]F_x=F \cos \theta_x \quad F_y=F \cos \theta_y \quad F_z=F \cos \theta_z[/latex] (2.19) [latex]\begin{array}{l}F_x=(500 \mathrm{~N}) \cos 60^{\circ}=+250 \mathrm{~N} \\F_y=(500 \mathrm{~N}) \cos 45^{\circ}=+354 \mathrm{~N} \\F_z=(500 \mathrm{~N}) \cos 120^{\circ}=-250 \mathrm{~N}\end{array}[/latex] Carrying these values into Eq. (2.20), you have [latex]\mathbf{F}=F_x \mathbf{i}+F_y \mathbf{j}+F_z \mathbf{k}[/latex] (2.20) [latex]\mathbf{F}=(250 \mathrm{~N}) \mathbf{i}+(354 \mathrm{~N}) \mathbf{j}-(250 \mathrm{~N}) \mathbf{k}[/latex] As in the case of two-dimensional problems, a plus sign indicates that the component has the same sense as the corresponding axis, and a minus sign indicates that it has the opposite sense.

Question 2.CA.4: A force of 500 N forms angles of 60°, 45°, and 120°, respect......

Vector Mechanics for Engineers Statics [1686415]

A force of 500 N forms angles of 60°, 45°, and 120°, respectively, with the x, y, and z axes. Find the components $F_x, F_y, \text { and } F_z$ of the force and express the force in terms of unit vectors.

Question Data is a breakdown of the data given in the question above.

Force: 500 N
Angles: 60°, 45°, and 120° with the x, y, and z axes respectively.

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

Step 1:

Substitute the given values for F, θx, θy, and θz into formulas (2.19). These formulas give the scalar components of F.

Step 2:

Calculate each component using the given values. Use the cosine function to find the components in each direction.

Step 3:

Plug in the calculated values of Fx, Fy, and Fz into equation (2.20). This equation gives the vector representation of F.

Step 4:

Write the final vector representation of F using the calculated values of Fx, Fy, and Fz. Use the unit vectors i, j, and k to indicate the direction of each component.

Step 5:

Explain the meaning of the plus and minus signs in the final vector representation. A plus sign indicates that the component has the same direction as the corresponding axis, while a minus sign indicates that it has the opposite direction.

Final Answer

Substitute $F=500 \mathrm{~N}, \theta_x=60^{\circ}, \theta_y=45^{\circ}, \text { and } \theta_z=120^{\circ}$ formulas (2.19). The scalar components of F are then

$F_x=F \cos \theta_x \quad F_y=F \cos \theta_y \quad F_z=F \cos \theta_z$ (2.19)

$\begin{array}{l}F_x=(500 \mathrm{~N}) \cos 60^{\circ}=+250 \mathrm{~N} \\F_y=(500 \mathrm{~N}) \cos 45^{\circ}=+354 \mathrm{~N} \\F_z=(500 \mathrm{~N}) \cos 120^{\circ}=-250 \mathrm{~N}\end{array}$

Carrying these values into Eq. (2.20), you have

$\mathbf{F}=F_x \mathbf{i}+F_y \mathbf{j}+F_z \mathbf{k}$ (2.20)

$\mathbf{F}=(250 \mathrm{~N}) \mathbf{i}+(354 \mathrm{~N}) \mathbf{j}-(250 \mathrm{~N}) \mathbf{k}$

As in the case of two-dimensional problems, a plus sign indicates that the component has the same sense as the corresponding axis, and a minus sign indicates that it has the opposite sense.