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The magnitude of a moment about a line segment connecting points Pand Q due to a force F applied at point R (with R not on the line through P and Q) can be calculated using the scalar triple product, Part A - Finding the scalar triple product Which of the following equations correctly evaluates the scalar triple product of three Cartesian vectors R, S, and T? MPQ = uPQ · r × F, • View Available Hint(s) where r is a position vector from any point on the line through P and Q to R and upo is the unit vector in the direction of line segment PQ. The unit vector uPQ is then multiplied by this magnitude to find the vector representation of the moment. O R.SxT= R,(S,T; – S;T,)+ R,(S„T; – S;T;) +R:(S„Ty – S,Tz) O R.S× T = R,(S,T; – S;T,) i+ R„(S,T; – S;T.) j+ R:(S,T, – S„T,) k O R.Sx T= R,(S,T; – S-T,) i – R,(S„T; – S;T2) j+R:(S,T, – S„T2) k O R.Sx T = R,(S„T; – S,T,) – R„(S„T; – S;T.) + R:(S„T, – S„T„) As shown in the figure, the member is anchored at A and section AB lies in the x-y plane. The dimensions are x1 = 1.6 m, Y1 = 1.7 m, and z1 = 1.6 m. The force applied at point C is F = [-185 i+ 80 j+ 190 k] N. Submit (Figure 1) Part B - Calculating the moment about AB using the position vector AC Using the position vector from A to C, calculate the moment about segment AB due to force F. Figure < 1 of 1> Express the individual components to three significant figures, if necessary, separated by commas. • View Available Hint(s) vec ? MAB =[ i, j, k] N · m B Submit

The magnitude of a moment about a line segment connecting points Pand Q due to a force F applied at point R (with R not on the line through P and Q) can be calculated using the scalar triple product, Part A - Finding the scalar triple product Which of the following equations correctly evaluates the scalar triple product of three Cartesian vectors R, S, and T? MPQ = uPQ · r × F, • View Available Hint(s) where r is a position vector from any point on the line through P and Q to R and upo is the unit vector in the direction of line segment PQ. The unit vector uPQ is then multiplied by this magnitude to find the vector representation of the moment. O R.SxT= R,(S,T; – S;T,)+ R,(S„T; – S;T;) +R:(S„Ty – S,Tz) O R.S× T = R,(S,T; – S;T,) i+ R„(S,T; – S;T.) j+ R:(S,T, – S„T,) k O R.Sx T= R,(S,T; – S-T,) i – R,(S„T; – S;T2) j+R:(S,T, – S„T2) k O R.Sx T = R,(S„T; – S,T,) – R„(S„T; – S;T.) + R:(S„T, – S„T„) As shown in the figure, the member is anchored at A and section AB lies in the x-y plane. The dimensions are x1 = 1.6 m, Y1 = 1.7 m, and z1 = 1.6 m. The force applied at point C is F = [-185 i+ 80 j+ 190 k] N. Submit (Figure 1) Part B - Calculating the moment about AB using the position vector AC Using the position vector from A to C, calculate the moment about segment AB due to force F. Figure < 1 of 1> Express the individual components to three significant figures, if necessary, separated by commas. • View Available Hint(s) vec ? MAB =[ i, j, k] N · m B Submit

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### Understanding the Magnitude of a Moment About a Line The magnitude of a moment about a line segment connecting points \( P \) and \( Q \) due to a force \( \mathbf{F} \) applied at point \( R \) (with \( R \) not on the line through \( P \) and \( Q \)) can be calculated using the scalar triple product: \[ M_{PQ} = \mathbf{u}_{PQ} \cdot (\mathbf{r} \times \mathbf{F}), \] where \(\mathbf{r}\) is a position vector from any point on the line through \( P \) and \( Q \) to \( R \) and \(\mathbf{u}_{PQ}\) is the unit vector in the direction of line segment \( PQ \). The unit vector \(\mathbf{u}_{PQ}\) is then multiplied by this magnitude to find the vector representation of the moment. As shown in the figure, the member is anchored at \( A \) and section \( AB \) lies in the \( x \text{-} y \) plane. The dimensions are \( x_1 = 1.6 \) m, \( y_1 = 1.7 \) m, and \( z_1 = 1.6 \) m. The force applied at point \( C \) is \[ \mathbf{F} = \left[ -185 \mathbf{i} + 80 \mathbf{j} + 190 \mathbf{k} \right] \text{ N}. \] (Figure 1) #### Figure Explanation The figure illustrates a 3D coordinate system with a member anchored at \( A \) and extending to point \( B \) in the \( x \text{-} y \) plane. From \( B \), the member extends vertically to point \( C \) where the force \(\mathbf{F}\) is applied. #### Part A - Finding the Scalar Triple Product **Question**: Which of the following equations correctly evaluates the scalar triple product of three Cartesian vectors \( \mathbf{R}, \ \mathbf{S}, \ \mathbf{T} \)? Choices: 1. \[ \mathbf{R} \cdot (\mathbf{S} \times \mathbf{T}) = R_x (S_y T_z - S_z T_y) + R_y (S_x T_z - S_z
Transcribed Image Text:### Understanding the Magnitude of a Moment About a Line The magnitude of a moment about a line segment connecting points \( P \) and \( Q \) due to a force \( \mathbf{F} \) applied at point \( R \) (with \( R \) not on the line through \( P \) and \( Q \)) can be calculated using the scalar triple product: \[ M_{PQ} = \mathbf{u}_{PQ} \cdot (\mathbf{r} \times \mathbf{F}), \] where \(\mathbf{r}\) is a position vector from any point on the line through \( P \) and \( Q \) to \( R \) and \(\mathbf{u}_{PQ}\) is the unit vector in the direction of line segment \( PQ \). The unit vector \(\mathbf{u}_{PQ}\) is then multiplied by this magnitude to find the vector representation of the moment. As shown in the figure, the member is anchored at \( A \) and section \( AB \) lies in the \( x \text{-} y \) plane. The dimensions are \( x_1 = 1.6 \) m, \( y_1 = 1.7 \) m, and \( z_1 = 1.6 \) m. The force applied at point \( C \) is \[ \mathbf{F} = \left[ -185 \mathbf{i} + 80 \mathbf{j} + 190 \mathbf{k} \right] \text{ N}. \] (Figure 1) #### Figure Explanation The figure illustrates a 3D coordinate system with a member anchored at \( A \) and extending to point \( B \) in the \( x \text{-} y \) plane. From \( B \), the member extends vertically to point \( C \) where the force \(\mathbf{F}\) is applied. #### Part A - Finding the Scalar Triple Product **Question**: Which of the following equations correctly evaluates the scalar triple product of three Cartesian vectors \( \mathbf{R}, \ \mathbf{S}, \ \mathbf{T} \)? Choices: 1. \[ \mathbf{R} \cdot (\mathbf{S} \times \mathbf{T}) = R_x (S_y T_z - S_z T_y) + R_y (S_x T_z - S_z
**Part C - Calculating the moment about AB using the position vector BC** Using the position vector from \( B \) to \( C \), calculate the moment about segment \( AB \) due to force \( \mathbf{F} \). **Express the individual components to three significant figures, if necessary, separated by commas.** View Available Hint(s) \[ M_{AB} = \left[ \ \ \ \ \ \ \ \ \ \ \ \ \ \right] \quad i, j, k \quad \text{N} \cdot \text{m} \] This part of the problem involves calculating the moment (torque) about a specific segment (AB) in a vector system where you know the position vector from point B to point C and the force \( \mathbf{F} \) that acts on it. Make sure to use proper vector operations and express your result as components in the \( i, j, \) and \( k \) directions, denoted in \(\text{N} \cdot \text{m} \). For hints or additional help, you can click on "View Available Hint(s)."
Transcribed Image Text:**Part C - Calculating the moment about AB using the position vector BC** Using the position vector from \( B \) to \( C \), calculate the moment about segment \( AB \) due to force \( \mathbf{F} \). **Express the individual components to three significant figures, if necessary, separated by commas.** View Available Hint(s) \[ M_{AB} = \left[ \ \ \ \ \ \ \ \ \ \ \ \ \ \right] \quad i, j, k \quad \text{N} \cdot \text{m} \] This part of the problem involves calculating the moment (torque) about a specific segment (AB) in a vector system where you know the position vector from point B to point C and the force \( \mathbf{F} \) that acts on it. Make sure to use proper vector operations and express your result as components in the \( i, j, \) and \( k \) directions, denoted in \(\text{N} \cdot \text{m} \). For hints or additional help, you can click on "View Available Hint(s)."
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