Question 1169042
To solve this using a scale diagram, you'll need paper, a ruler, and a protractor. Here's a step-by-step guide on how to sketch and determine the resultant vector for each case:

**General Procedure for Scale Diagrams:**

1.  **Choose a suitable scale:** Select a scale that allows you to represent the given magnitudes on your paper. For example, 1 cm on your diagram could represent 1 m or 1 km, depending on the problem.
2.  **Draw the first vector:** Starting from an origin, draw the first vector to scale in the correct direction. Use your ruler to measure the length according to your chosen scale and your protractor to ensure the correct angle (if any). Draw an arrowhead at the end to indicate the direction.
3.  **Draw the second vector:** Starting from the arrowhead of the first vector, draw the second vector to scale in its correct direction. Again, use your ruler and protractor for accuracy.
4.  **Draw the resultant vector:** The resultant vector starts from the origin (the tail of the first vector) and ends at the arrowhead of the last vector drawn.
5.  **Measure the resultant vector:**
    * **Magnitude:** Use your ruler to measure the length of the resultant vector. Convert this length back to the original units using your chosen scale.
    * **Direction:** Use your protractor to measure the angle of the resultant vector relative to a reference direction (e.g., North, East, or a horizontal line). Express the direction clearly.

**a) 7 m (down) and 9 m (left)**

1.  **Choose a scale:** Let 1 cm on the paper represent 1 m.
2.  **Draw the first vector:** Draw a vertical line 7 cm long pointing downwards from your chosen origin. Label it $\vec{A}$.
3.  **Draw the second vector:** Starting from the bottom of the first vector, draw a horizontal line 9 cm long pointing to the left. Label it $\vec{B}$.
4.  **Draw the resultant vector:** Draw a straight line from the origin (start of $\vec{A}$) to the end of $\vec{B}$. Label it $\vec{R}$.
5.  **Measure the resultant vector:**
    * **Magnitude:** Measure the length of $\vec{R}$ with your ruler. It should be approximately 11.4 cm. Using the scale, the magnitude is approximately **11.4 m**.
    * **Direction:** Measure the angle of $\vec{R}$ with respect to the downward vertical line (North in a standard compass, if we consider down as South and left as West). The angle to the left of the downward line should be approximately $52^\circ$. Therefore, the direction is approximately **$52^\circ$ West of South** [S $52^\circ$ W]. Alternatively, the angle below the leftward horizontal line (West) should be approximately $38^\circ$, so the direction is **$38^\circ$ South of West** [W $38^\circ$ S].

    **Sketch:**

    ```
        Start (Origin)
        |
        | 7 cm (Down)  (A)
        v
        +------- 9 cm (Left) (B)
                <-----o (End)
                \   R (Resultant)
                 \  ~11.4 cm
                  \
                   Angle ~ 52° (with down) or ~ 38° (with left)
    ```

**b) 6 km (N55°W) and 4 km (E35°S)**

1.  **Choose a scale:** Let 1 cm on the paper represent 1 km.
2.  **Draw the first vector:**
    * Draw a North-South and East-West axis on your paper.
    * Starting from the origin, draw a line 6 cm long at an angle of $55^\circ$ West of North. This means you measure $55^\circ$ from the North line towards the West. Label it $\vec{A}$.
3.  **Draw the second vector:**
    * Starting from the arrowhead of the first vector, draw a line 4 cm long at an angle of $35^\circ$ South of East. This means you measure $35^\circ$ from the East line towards the South. Label it $\vec{B}$.
4.  **Draw the resultant vector:** Draw a straight line from the origin (start of $\vec{A}$) to the end of $\vec{B}$. Label it $\vec{R}$.
5.  **Measure the resultant vector:**
    * **Magnitude:** Measure the length of $\vec{R}$ with your ruler. It should be approximately 5.1 cm. Using the scale, the magnitude is approximately **5.1 km**.
    * **Direction:** Measure the angle of $\vec{R}$ relative to the North or East axis at the origin. You'll find the angle is approximately $78^\circ$ West of North [N $78^\circ$ W].

    **Sketch:**

    ```
            N
            |
            | 6 cm (N55°W) (A)
           / \
          /   \
         o-----+--- E
        Start    \
                 \  4 cm (E35°S) (B)
                  \ /
                   v
                   o (End)
                   \  R (Resultant) ~ 5.1 cm
                    \
                     Angle ~ 78° (with North)
    ```

**Important Note:** The accuracy of your results depends heavily on the precision of your drawing and measurements. Using a sharp pencil, a precise ruler, and an accurate protractor is crucial for obtaining reliable answers with the scale diagram method. The numerical answers obtained through calculation (as done in the previous response) will generally be more precise than those from a scale diagram.