The vectors `veca and vecb` are not perpendicular and `vecac and vecd` are two vectors satisfying : `vecbxxvecc=vecbxxvecd and veca.vecd=0.` Then the `vecd` is equal to (A) `vecc+(veca.vecc)/(veca.vecb))vecb` (B) `vecb+(vecb.vecc)/(veca.vecb))vecc` (C) `vecc-(veca.vecc)/(veca.vecb))vecb` (D) `vecb-(vecb.vecc)/(veca.vecb))vecc`

Options

A. `1/2(3a^(2) + b^(2)-c^(2))`
B. `vecb.((vecb.vecc)/(veca.vecb))vecc`
C. `vecc = ((veca.vecc)/(veca.vecb))vecb`
D. `vecb - (vecb.vecc)/(veca.vecb)vecc`

Answer

To solve the problem step by step, we need to analyze the given conditions and apply vector identities accordingly.

### Step 1: Understand the Given Conditions
We have two vectors \( \vec{a} \) and \( \vec{b} \) that are not perpendicular. We also have two vectors \( \vec{c} \) and \( \vec{d} \) such that:
1. \( \vec{b} \times \vec{c} = \vec{b} \times \vec{d} \)
2. \( \vec{a} \cdot \vec{d} = 0 \)

### Step 2: Analyze the Cross Product Condition
From the equation \( \vec{b} \times \vec{c} = \vec{b} \times \vec{d} \), we can use the property of cross products. This implies that the vector \( \vec{b} \) is perpendicular to the vector \( \vec{c} - \vec{d} \):
\[ \vec{b} \times (\vec{c} - \vec{d}) = \vec{0} \]
This means \( \vec{c} - \vec{d} \) is parallel to \( \vec{b} \). Hence, we can express \( \vec{d} \) in terms of \( \vec{c} \):
\[ \vec{d} = \vec{c} + k \vec{b} \]
for some scalar \( k \).

### Step 3: Use the Dot Product Condition
Next, we utilize the condition \( \vec{a} \cdot \vec{d} = 0 \):
\[ \vec{a} \cdot (\vec{c} + k \vec{b}) = 0 \]
This expands to:
\[ \vec{a} \cdot \vec{c} + k (\vec{a} \cdot \vec{b}) = 0 \]
From this, we can solve for \( k \):
\[ k = -\frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}} \]

### Step 4: Substitute Back to Find \( \vec{d} \)
Now, substituting \( k \) back into the equation for \( \vec{d} \):
\[ \vec{d} = \vec{c} - \frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}} \vec{b} \]

### Step 5: Final Expression for \( \vec{d} \)
Thus, we have:
\[ \vec{d} = \vec{c} - \frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}} \vec{b} \]
This matches with option (C):
\[ \vec{d} = \vec{c} - \left( \frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}} \right) \vec{b} \]

### Conclusion
The correct answer is option (C).

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