If a, b , c are vectors such that `c = a + b` and a. b = 0 , then

Options

A. `a^(2) + b^(2) + c^(2) = 0`
B. `a^(2) - b^(2) = 0`
C. `a^(2) + b^(2) = c^(2)`
D. `c = a xx b`

Answer

To solve the problem, we need to analyze the given information about the vectors \( a \), \( b \), and \( c \). We know that:

1. \( c = a + b \)
2. \( a \cdot b = 0 \)

We need to find out what can be concluded from these statements.

### Step-by-step Solution:

**Step 1: Understanding the Dot Product Condition**

Since \( a \cdot b = 0 \), it implies that vectors \( a \) and \( b \) are perpendicular to each other. This is a key property of the dot product: if the dot product of two vectors is zero, then the vectors are orthogonal.

**Hint for Step 1:** Recall that the dot product of two vectors is zero if and only if they are perpendicular.

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**Step 2: Expressing \( c \) in Terms of \( a \) and \( b \)**

We have the equation \( c = a + b \). To find the magnitude of \( c \), we can square both sides:

\[ |c|^2 = |a + b|^2 \]

**Hint for Step 2:** Remember that the magnitude of a vector squared can be expressed using the dot product.

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**Step 3: Expanding the Magnitude Squared**

Using the property of the dot product, we can expand the right-hand side:

\[ |c|^2 = (a + b) \cdot (a + b) = a \cdot a + 2(a \cdot b) + b \cdot b \]

**Hint for Step 3:** When expanding, remember to apply the distributive property of the dot product.

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**Step 4: Substituting the Dot Product Values**

Since we know \( a \cdot b = 0 \), we can substitute this into our equation:

\[ |c|^2 = |a|^2 + 2(0) + |b|^2 = |a|^2 + |b|^2 \]

**Hint for Step 4:** Simplify the equation by substituting known values.

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**Step 5: Conclusion**

From the above equation, we conclude that:

\[ |c|^2 = |a|^2 + |b|^2 \]

This is the Pythagorean theorem, which holds true because \( a \) and \( b \) are perpendicular. Therefore, we can say that:

\[ |c| = \sqrt{|a|^2 + |b|^2} \]

This means that the magnitude of vector \( c \) is equal to the square root of the sum of the squares of the magnitudes of vectors \( a \) and \( b \).

**Final Answer:** The correct conclusion is that \( |c|^2 = |a|^2 + |b|^2 \).

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