If `|vec(a)+vec(b)|ge|vec(a)-vec(b)|` then angle between `vec(a)` and `vec(b)` may be

Options

A. `50^(@)`
B. `90^(@)`
C. `60^(@)`
D. `120^(@)`

Explanation

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

\[ |\vec{a} + \vec{b}| \geq |\vec{a} - \vec{b}| \]

### Step 1: Square both sides of the inequality

Since both sides are non-negative, we can square them without changing the inequality:

\[ |\vec{a} + \vec{b}|^2 \geq |\vec{a} - \vec{b}|^2 \]

### Step 2: Expand both sides using the formula for the square of a vector's magnitude

Using the property that \( |\vec{x}|^2 = \vec{x} \cdot \vec{x} \), we expand both sides:

\[ (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) \geq (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b}) \]

This gives us:

\[ |\vec{a}|^2 + 2\vec{a} \cdot \vec{b} + |\vec{b}|^2 \geq |\vec{a}|^2 - 2\vec{a} \cdot \vec{b} + |\vec{b}|^2 \]

### Step 3: Simplify the inequality

Now, we can simplify the inequality by canceling out the common terms:

\[ |\vec{a}|^2 + 2\vec{a} \cdot \vec{b} + |\vec{b}|^2 - |\vec{a}|^2 - |\vec{b}|^2 \geq -2\vec{a} \cdot \vec{b} \]

This simplifies to:

\[ 2\vec{a} \cdot \vec{b} \geq -2\vec{a} \cdot \vec{b} \]

### Step 4: Combine like terms

Combining the terms gives us:

\[ 4\vec{a} \cdot \vec{b} \geq 0 \]

### Step 5: Divide by 4

Dividing both sides by 4 (which is positive) does not change the inequality:

\[ \vec{a} \cdot \vec{b} \geq 0 \]

### Step 6: Relate the dot product to the angle

The dot product can be expressed in terms of the angle \( \theta \) between the vectors:

\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]

Thus, we have:

\[ |\vec{a}| |\vec{b}| \cos \theta \geq 0 \]

Since \( |\vec{a}| \) and \( |\vec{b}| \) are both positive (assuming non-zero vectors), we can conclude that:

\[ \cos \theta \geq 0 \]

### Step 7: Determine the angle range

The condition \( \cos \theta \geq 0 \) implies that the angle \( \theta \) between the vectors \( \vec{a} \) and \( \vec{b} \) can be:

\[ 0^\circ \leq \theta \leq 90^\circ \]

### Final Conclusion

The angle between \( \vec{a} \) and \( \vec{b} \) may be between \( 0^\circ \) and \( 90^\circ \) inclusive.

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