The x-component of the resultant of several vectors

Options

A. is equal to the sum of the x-components of the vectors
B. may be smaller than the sum of the magnitudes of the vectors
C. may be greater than the sum of the magnitude of the vectors
D. may be equal to the sum of the magnitude of the vectors.

Answer

To solve the question regarding the x-component of the resultant of several vectors, we will analyze the options provided step by step.

### Step 1: Understanding the x-component of the resultant vector
The x-component of the resultant vector from multiple vectors is determined by summing the x-components of each individual vector.

### Step 2: Analyzing Option 1
**Option 1 states:** "The x-component of the resultant vector is equal to the sum of the x-components of the vectors."
- If we have vectors A and B, with components \( A_x \) and \( B_x \) respectively, the x-component of the resultant vector \( R_x \) can be expressed as:
\[ R_x = A_x + B_x \]
- This is true for any number of vectors. Thus, **Option 1 is correct.**

### Step 3: Analyzing Option 2
**Option 2 states:** "The x-component of the resultant vector may be smaller than the sum of the magnitudes of the vectors."
- The magnitude of a vector is given by the square root of the sum of the squares of its components. Hence, the magnitude of vector A is:
\[ |A| = \sqrt{A_x^2 + A_y^2} \]
- The sum of the magnitudes of the vectors can be greater than the x-component of the resultant if the vectors are not aligned in the same direction. Thus, **Option 2 is correct.**

### Step 4: Analyzing Option 3
**Option 3 states:** "The x-component of the resultant vector may be greater than the sum of the magnitudes of the vectors."
- This statement is false because the x-component of the resultant can never exceed the sum of the magnitudes of the vectors. The x-component is just one part of the vector, while the magnitude considers all components. Thus, **Option 3 is incorrect.**

### Step 5: Analyzing Option 4
**Option 4 states:** "The x-component of the resultant vector may be equal to the sum of the magnitudes of the vectors."
- This can occur in a special case where all vectors are aligned along the x-axis. For example, if vector A has a magnitude of 3 and vector B has a magnitude of 4, and both are in the positive x-direction, then:
\[ R_x = A + B = 3 + 4 = 7 \]
- In this case, the x-component of the resultant vector equals the sum of the magnitudes. Thus, **Option 4 is correct.**

### Conclusion
The correct options are:
- **Option 1:** Correct
- **Option 2:** Correct
- **Option 3:** Incorrect
- **Option 4:** Correct

### Summary of Correct Options
The correct statements regarding the x-component of the resultant vector are Options 1, 2, and 4.

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