If a, b and c are non-coplanar vectors and `d=lambdaa+mub+nuc`, then `lambda` is equal to

Options

A. `([d b c])/([b a c])`
B. `([b c d])/([b c a])`
C. `([b d c])/([a b c])`
D. `([c b d])/([a b c])`

Answer

To find the value of \( \lambda \) in the equation \( \mathbf{d} = \lambda \mathbf{a} + \mu \mathbf{b} + \nu \mathbf{c} \), where \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are non-coplanar vectors, we can follow these steps:

### Step 1: Write the given equation
We start with the equation:
\[ \mathbf{d} = \lambda \mathbf{a} + \mu \mathbf{b} + \nu \mathbf{c} \]

### Step 2: Take the dot product with \( \mathbf{b} \times \mathbf{c} \)
Next, we take the dot product of both sides of the equation with \( \mathbf{b} \times \mathbf{c} \):
\[ \mathbf{d} \cdot (\mathbf{b} \times \mathbf{c}) = (\lambda \mathbf{a} + \mu \mathbf{b} + \nu \mathbf{c}) \cdot (\mathbf{b} \times \mathbf{c}) \]

### Step 3: Simplify the right-hand side
Using the properties of the dot product, we can simplify the right-hand side:
\[ \mathbf{d} \cdot (\mathbf{b} \times \mathbf{c}) = \lambda (\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})) + \mu (\mathbf{b} \cdot (\mathbf{b} \times \mathbf{c})) + \nu (\mathbf{c} \cdot (\mathbf{b} \times \mathbf{c})) \]
Since \( \mathbf{b} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \) and \( \mathbf{c} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \), we have:
\[ \mathbf{d} \cdot (\mathbf{b} \times \mathbf{c}) = \lambda (\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})) \]

### Step 4: Solve for \( \lambda \)
Now, we can isolate \( \lambda \):
\[ \lambda = \frac{\mathbf{d} \cdot (\mathbf{b} \times \mathbf{c})}{\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})} \]

### Step 5: Reorganize the expression
We can express this in terms of scalar triple products. The scalar triple product \( [\mathbf{a}, \mathbf{b}, \mathbf{c}] \) can be represented as:
\[ \lambda = \frac{[\mathbf{d}, \mathbf{b}, \mathbf{c}]}{[\mathbf{a}, \mathbf{b}, \mathbf{c}]} \]

### Conclusion
Thus, the final expression for \( \lambda \) is:
\[ \lambda = \frac{[\mathbf{b}, \mathbf{c}, \mathbf{d}]}{[\mathbf{b}, \mathbf{c}, \mathbf{a}]} \]