# Solution Explanation Let's address each part as described in the problem and based on the diagram. ## i. **Vector \(\vec{W}\)** - **Condition:** \(\vec{W}\) has three times the magnitude and is in the same direction as \(\vec{A}\). - **Action:** Draw a vector starting from the origin, pointing in the same direction as \(\vec{A}\), but three times longer. --- ## ii. **Vector \(\vec{X}\)** - **Condition:** \(\vec{X}\) is the resultant force of \(\vec{B}\) and \(\vec{C}\). - **Action:** Add \(\vec{B}\) and \(\vec{C}\) using the head-to-tail method. Draw \(\vec{C}\) starting at the tip of \(\vec{B}\); \(\vec{X}\) goes from the origin to the tip of \(\vec{C}\). --- ## iii. **Vector \(\vec{Y}\)** - **Condition:** \(\vec{Y}\) is the resultant force of \(\vec{A}\) and \(\vec{B}\). - **Action:** Add \(\vec{A}\) and \(\vec{B}\) using the head-to-tail method. Draw \(\vec{A}\) first; from its tip, draw \(\vec{B}\). Then, \(\vec{Y}\) goes from the origin to the tip of \(\vec{B}\). --- ## iv. **Vector \(\vec{Z}\)** - **Condition:** \(\vec{Z}\) is the equilibrant force of \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\). - **Action:** The equilibrant is equal in magnitude but opposite in direction to the resultant of \(\vec{A}+\vec{B}+\vec{C}\). --- ## **Annotated Diagram** Below is a description of where each vector should be drawn: #### 1. **\(\vec{W}\)** - Start at origin, three times the length of \(\vec{A}\), same direction as \(\vec{A}\). #### 2. **\(\vec{X} = \vec{B} + \vec{C}\)** - Using head-to-tail addition, draw \(\vec{C}\) starting from the tip of \(\vec{B}\). Then, from the origin to the tip of \(\vec{C}\) is \(\vec{X}\). #### 3. **\(\vec{Y} = \vec{A} + \vec{B}\)** - Draw \(\vec{B}\) starting from the tip of \(\vec{A}\). From the origin to the tip of this new vector is \(\vec{Y}\). #### 4. **\(\vec{Z}\) (Equilibrant)** - Find the resultant: \(\vec{R} = \vec{A} + \vec{B} + \vec{C}\). - \(\vec{Z}\) is equal in magnitude but points in the opposite direction to \(\vec{R}\). --- ## **Summary Table** | Vector | Description | |--------|-------------| | \(\vec{W}\) | Three times the length of \(\vec{A}\), same direction | | \(\vec{X}\) | Resultant of \(\vec{B} + \vec{C}\) | | \(\vec{Y}\) | Resultant of \(\vec{A} + \vec{B}\) | | \(\vec{Z}\) | Opposite to resultant of (\(\vec{A} + \vec{B} + \vec{C}\)) | --- **If you would like me to illustrate the diagram with the vectors added, please let me know!**

# Vector Forces Illustration Here's a visual representation of the vector forces as described in your problem. ## Diagram of Vectors  *Alt text: Diagram illustrating the vector forces W, X, Y, and Z in relation to vectors A, B, and C.* ## Vector Descriptions ### 1. **Vector \(\vec{W}\)** - **Description:** Three times the magnitude of \(\vec{A}\) in the same direction. - **Drawing:** Start at the origin, extend in the direction of \(\vec{A}\). ### 2. **Vector \(\vec{X}\)** - **Description:** Resultant of vectors \(\vec{B}\) and \(\vec{C}\). - **Drawing:** Use head-to-tail method. Draw \(\vec{C}\) starting from the tip of \(\vec{B}\) and extend to find \(\vec{X}\). ### 3. **Vector \(\vec{Y}\)** - **Description:** Resultant of vectors \(\vec{A}\) and \(\vec{B}\). - **Drawing:** Draw \(\vec{B}\) from the tip of \(\vec{A}\). The resultant \(\vec{Y}\) goes from the origin to the tip of \(\vec{B}\). ### 4. **Vector \(\vec{Z}\)** - **Description:** Equilibrant to the resultant of \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\). - **Drawing:** Calculate the resultant vector \(\vec{R} = \vec{A} + \vec{B} + \vec{C}\), then draw \(\vec{Z}\) in the opposite direction. ## Summary Table of Vectors | Vector | Description | |--------|-------------| | \(\vec{W}\) | Three times the length of \(\vec{A}\), same direction | | \(\vec{X}\) | Resultant of \(\vec{B} + \vec{C}\) | | \(\vec{Y}\) | Resultant of \(\vec{A} + \vec{B}\) | | \(\vec{Z}\) | Opposite to resultant of (\(\vec{A} + \vec{B} + \vec{C}\)) | This structured approach provides clarity on how to visualize and calculate each vector's placement in relation to the others.