If `veca, vecb, vecc` are vectors such that `veca.vecb=0` and `veca + vecb = vecc` then:

Options

A. `|veca|^(2) + |vecb|^(2) = |vecc|^(2)`
B. `|veca|^(2) = |vecb|^(2) + |vecc|^(2)`
C. `|vecb|^(2) = |veca|^(2) + |vecc|^(2)`
D. None of these

Answer

To solve the problem step by step, we start with the given conditions about the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).

### Step 1: Understand the Given Conditions
We know:
1. \(\vec{a} \cdot \vec{b} = 0\) (This means that vectors \(\vec{a}\) and \(\vec{b}\) are perpendicular.)
2. \(\vec{a} + \vec{b} = \vec{c}\)

### Step 2: Square Both Sides of the Equation
We will square the equation \(\vec{a} + \vec{b} = \vec{c}\):
\[ (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = \vec{c} \cdot \vec{c} \]

### Step 3: Expand the Left Side
Using the distributive property of the dot product, we expand the left side:
\[ \vec{a} \cdot \vec{a} + 2(\vec{a} \cdot \vec{b}) + \vec{b} \cdot \vec{b} = \vec{c} \cdot \vec{c} \]

### Step 4: Substitute the Known Values
Since we know that \(\vec{a} \cdot \vec{b} = 0\), we can substitute this into our equation:
\[ \vec{a} \cdot \vec{a} + 2(0) + \vec{b} \cdot \vec{b} = \vec{c} \cdot \vec{c} \]
This simplifies to:
\[ \vec{a} \cdot \vec{a} + \vec{b} \cdot \vec{b} = \vec{c} \cdot \vec{c} \]

### Step 5: Rewrite in Terms of Magnitudes
We can express the dot products in terms of magnitudes:
\[ |\vec{a}|^2 + |\vec{b}|^2 = |\vec{c}|^2 \]

### Conclusion
Thus, we have derived the relationship:
\[ |\vec{a}|^2 + |\vec{b}|^2 = |\vec{c}|^2 \]
This means that the magnitude of vector \(\vec{c}\) is equal to the square root of the sum of the squares of the magnitudes of vectors \(\vec{a}\) and \(\vec{b}\).

### Final Answer
The condition that holds true is:
\[ |\vec{a}|^2 + |\vec{b}|^2 = |\vec{c}|^2 \]

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