The vectors `vec(a), vec(b), vec(c ) and vec(d)` are such that `vec(a) xx vec(b) = vec(c ) xx vec(d) and vec(a) xx vec(c )= vec(b) xx vec(d)`. Which of the following is/are correct?
1. `(vec(a)- vec(d)) xx (vec(b) - vec(c ))= vec(0)`
2. `(vec(a) xx vec(b))xx (vec(c ) xx vec(d))= vec(0)`
Select the correct answer using the codes given below

Options

A. A. Only 1
B. B. Only 2
C. C. Both 1 and 2
D. D. Neither 1 nor 2

Answer

To solve the problem, we need to analyze the given conditions and verify the two statements one by one.

### Given:
1. \( \vec{a} \times \vec{b} = \vec{c} \times \vec{d} \) (Equation 1)
2. \( \vec{a} \times \vec{c} = \vec{b} \times \vec{d} \) (Equation 2)

### Statement 1:
We need to check if \( (\vec{a} - \vec{d}) \times (\vec{b} - \vec{c}) = \vec{0} \).

#### Step 1: Expand the left-hand side
Using the distributive property of the cross product:
\[ (\vec{a} - \vec{d}) \times (\vec{b} - \vec{c}) = \vec{a} \times \vec{b} - \vec{a} \times \vec{c} - \vec{d} \times \vec{b} + \vec{d} \times \vec{c} \]

#### Step 2: Substitute using the given equations
From Equation 1, we know \( \vec{a} \times \vec{b} = \vec{c} \times \vec{d} \). Thus, we can replace \( \vec{a} \times \vec{b} \) with \( \vec{c} \times \vec{d} \):
\[ = (\vec{c} \times \vec{d}) - \vec{a} \times \vec{c} - \vec{d} \times \vec{b} + \vec{d} \times \vec{c} \]

From Equation 2, we know \( \vec{a} \times \vec{c} = \vec{b} \times \vec{d} \). We can replace \( \vec{a} \times \vec{c} \) with \( \vec{b} \times \vec{d} \):
\[ = (\vec{c} \times \vec{d}) - (\vec{b} \times \vec{d}) - \vec{d} \times \vec{b} + \vec{d} \times \vec{c} \]

#### Step 3: Simplify the expression
Now, observe that \( -\vec{b} \times \vec{d} - \vec{d} \times \vec{b} = -(\vec{b} \times \vec{d} + \vec{d} \times \vec{b}) = 0 \) because the cross product is anti-commutative:
\[ = \vec{c} \times \vec{d} + \vec{d} \times \vec{c} = 0 \]

Thus, we conclude that:
\[ (\vec{a} - \vec{d}) \times (\vec{b} - \vec{c}) = \vec{0} \]

### Statement 1 is true.

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### Statement 2:
We need to check if \( (\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) = \vec{0} \).

#### Step 1: Substitute using the given equations
From Equation 1, we know \( \vec{c} \times \vec{d} = \vec{a} \times \vec{b} \). Therefore, we can rewrite the expression:
\[ (\vec{a} \times \vec{b}) \times (\vec{a} \times \vec{b}) \]

#### Step 2: Use the property of cross products
The cross product of any vector with itself is zero:
\[ \vec{a} \times \vec{a} = \vec{0} \]

Thus:
\[ (\vec{a} \times \vec{b}) \times (\vec{a} \times \vec{b}) = \vec{0} \]

### Statement 2 is also true.

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### Conclusion:
Both statements are true. Therefore, the correct answer is that both statements are correct.

### Final Answer:
Both Statement 1 and Statement 2 are correct.

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