Let the angle between two non-zero vectors `vec(A)` and `vec(B)` be `120^(@)` and its resultant be `vec(C)` then:

Options

A. `|vec(C)|`must be equal `||vec(A)|-|vec(B) ||`
B. `|vec(C)|`must be less than `||vec(A)|-|vec(B)||`
C. `|vec(C)|` must be greater than `||vec(A)|-|vec(B)||`
D. `|vec(C)|` may be equal to `||vec(A)|-|vec(B)||`

Answer

To find the resultant vector \( \vec{C} \) of two non-zero vectors \( \vec{A} \) and \( \vec{B} \) that form an angle of \( 120^\circ \), we can use the law of cosines. Here’s a step-by-step solution:

### Step 1: Understand the Law of Cosines
The law of cosines states that for any two vectors \( \vec{A} \) and \( \vec{B} \) forming an angle \( \theta \), the magnitude of the resultant vector \( \vec{C} \) can be calculated using the formula:
\[ |\vec{C}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \cos(\theta)} \]

### Step 2: Substitute the Given Values
In this case, the angle \( \theta = 120^\circ \). The cosine of \( 120^\circ \) is:
\[ \cos(120^\circ) = -\frac{1}{2} \]
Substituting this into the formula gives:
\[ |\vec{C}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \left(-\frac{1}{2}\right)} \]

### Step 3: Simplify the Expression
Now simplify the expression:
\[ |\vec{C}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 - |\vec{A}| |\vec{B}|} \]

### Step 4: Analyze the Resultant Magnitude
From the expression \( |\vec{C}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 - |\vec{A}| |\vec{B}|} \), we can see that the magnitude of \( \vec{C} \) depends on the magnitudes of \( \vec{A} \) and \( \vec{B} \).

### Step 5: Compare with Other Options
1. **Magnitude of \( \vec{C} \) must be equal to \( |\vec{A}| - |\vec{B}| \)**: This is incorrect because the resultant is not simply the difference of the magnitudes.
2. **Magnitude of \( \vec{C} \) must be less than \( |\vec{A}| + |\vec{B}| \)**: This is true, but we need to determine if it can be equal or greater.
3. **Magnitude of \( \vec{C} \) must be greater than \( ||\vec{A}| - |\vec{B}|| \)**: This is correct because the resultant will always be greater than the absolute difference of the two vectors when the angle is obtuse (greater than \( 90^\circ \)).
4. **Magnitude of \( \vec{C} \) may be equal to \( |\vec{A}| + |\vec{B}| \)**: This is incorrect because the angle is \( 120^\circ \), which means they are not aligned.

### Conclusion
Thus, the correct statement about the resultant vector \( \vec{C} \) is that its magnitude must be greater than the absolute difference of the magnitudes of \( \vec{A} \) and \( \vec{B} \).

### Final Answer
The answer is that the magnitude of \( \vec{C} \) must be greater than \( ||\vec{A}| - |\vec{B}|| \).

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