It is found that `|A+B|=|A|`,This necessarily implies.

Options

A. B=0
B. A, B are antiparallel
C. A, B are perpendicular
D. `A.B le 0`

Explanation

To solve the problem, we need to analyze the given condition:

**Given:** \(|A + B| = |A|\)

This implies that the magnitude of the vector sum of \(A\) and \(B\) is equal to the magnitude of vector \(A\).

### Step 1: Understand the Magnitude Condition
We start from the equation:
\[ |A + B| = |A| \]
This means that the length of the vector \(A + B\) is equal to the length of vector \(A\).

### Step 2: Use the Parallelogram Law of Vector Addition
According to the parallelogram law, the magnitude of the sum of two vectors can be expressed as:
\[ |A + B|^2 = |A|^2 + |B|^2 + 2|A||B|\cos(\theta) \]
where \(\theta\) is the angle between vectors \(A\) and \(B\).

### Step 3: Substitute the Given Condition
Substituting the given condition into the equation:
\[ |A|^2 = |A|^2 + |B|^2 + 2|A||B|\cos(\theta) \]
Since \(|A|^2\) appears on both sides, we can cancel it out:
\[ 0 = |B|^2 + 2|A||B|\cos(\theta) \]

### Step 4: Rearranging the Equation
Rearranging gives us:
\[ |B|^2 + 2|A||B|\cos(\theta) = 0 \]

### Step 5: Analyze the Implications
This equation can be satisfied if:
1. \(|B| = 0\) (which means vector \(B\) is the zero vector).
2. The term \(2|A||B|\cos(\theta) = -|B|^2\) must hold true. This implies that if \(|B| \neq 0\), then:
\[ \cos(\theta) = -\frac{|B|}{2|A|} \]
This indicates that \(B\) and \(A\) could be anti-parallel if \(|B| = 2|A|\) and \(\theta = 180^\circ\).

### Step 6: Conclusion
From the analysis, we conclude that:
- Either \(B\) is the zero vector, or
- \(A\) and \(B\) are anti-parallel.

Thus, the correct implications of the condition \(|A + B| = |A|\) are:
1. \(|B| = 0\)
2. \(A\) and \(B\) are anti-parallel.

### Final Answer
The necessary implications of \(|A + B| = |A|\) are:
1. \(B = 0\)
2. \(A\) and \(B\) are anti-parallel.

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