If `(veca xx vecb) xx (vec c xx vecd)=h veca+k vecb=r vec c+s vecd`, where `veca, vecb` are non-collinear and `vec c, vec d` are also non-collinear then :

Options

A. `h=[vecb vec c vecd]`
B. `k=[veca vec c vecd]`
C. `r=[veca vecb vecd]`
D. `s= -[veca vecb vec c]`

Answer

To solve the equation \((\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) = h \vec{a} + k \vec{b} = r \vec{c} + s \vec{d}\), we will use the properties of vector products, specifically the vector triple product.

### Step 1: Define the Cross Products
Let us define:
\[ \vec{p} = \vec{a} \times \vec{b} \]
Thus, we can rewrite our equation as:
\[ \vec{p} \times (\vec{c} \times \vec{d}) = h \vec{a} + k \vec{b} \]

### Step 2: Apply the Vector Triple Product Identity
Using the vector triple product identity:
\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]
we can apply it to our equation:
\[ \vec{p} \times (\vec{c} \times \vec{d}) = (\vec{p} \cdot \vec{d}) \vec{c} - (\vec{p} \cdot \vec{c}) \vec{d} \]

### Step 3: Substitute \(\vec{p}\)
Substituting \(\vec{p} = \vec{a} \times \vec{b}\) into the equation:
\[ ((\vec{a} \times \vec{b}) \cdot \vec{d}) \vec{c} - ((\vec{a} \times \vec{b}) \cdot \vec{c}) \vec{d} = h \vec{a} + k \vec{b} \]

### Step 4: Compare Coefficients
Now we can compare coefficients on both sides of the equation. The left side gives us:
- Coefficient of \(\vec{c}\): \((\vec{a} \times \vec{b}) \cdot \vec{d}\)
- Coefficient of \(\vec{d}\): \(-(\vec{a} \times \vec{b}) \cdot \vec{c}\)

From the right side, we have:
- Coefficient of \(\vec{a}\): \(h\)
- Coefficient of \(\vec{b}\): \(k\)

### Step 5: Write the Values of \(r\) and \(s\)
Now, we can express the coefficients:
- For \(r\) and \(s\), we can also express the equation in terms of \(\vec{c}\) and \(\vec{d}\):
Let \(\vec{q} = \vec{c} \times \vec{d}\):
\[ \vec{a} \times \vec{b} \times \vec{q} = r \vec{c} + s \vec{d} \]
Applying the vector triple product identity again:
\[ (\vec{a} \times \vec{b}) \cdot \vec{d} \vec{c} - (\vec{a} \times \vec{b}) \cdot \vec{c} \vec{d} = r \vec{c} + s \vec{d} \]

### Step 6: Final Values
From the above comparisons, we can conclude:
- \(h = (\vec{a} \times \vec{b}) \cdot \vec{d}\)
- \(k = -(\vec{a} \times \vec{b}) \cdot \vec{c}\)
- \(r = (\vec{a} \times \vec{b}) \cdot \vec{d}\)
- \(s = -(\vec{a} \times \vec{b}) \cdot \vec{c}\)

### Summary of Values
Thus, we have:
- \(h = (\vec{a} \times \vec{b}) \cdot \vec{d}\)
- \(k = -(\vec{a} \times \vec{b}) \cdot \vec{c}\)
- \(r = (\vec{a} \times \vec{b}) \cdot \vec{d}\)
- \(s = -(\vec{a} \times \vec{b}) \cdot \vec{c}\)