If `veca, vecb, vecc` are non coplanar vectors and `lamda` is a real number, then `[(lamda(veca+vecb), lamda^(2)vecb, lamdavecc)]=[(veca, vecb+vecc,vecb)]` for

Options

A. exactly two values of `lamda`
B. exactly one value of `lamda`
C. exactly three values of `lamda`.
D. no value of `lamda`

Answer

To solve the problem, we need to analyze the given equation involving the box product of vectors. The box product is defined as the scalar triple product of three vectors. We will denote the vectors as \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), and we are given the equation:

\[ [\lambda(\vec{a} + \vec{b}), \lambda^2 \vec{b}, \lambda \vec{c}] = [\vec{a}, \vec{b} + \vec{c}, \vec{b}] \]

### Step 1: Rewrite the Box Product
The box product can be expressed as:

\[ [\vec{x}, \vec{y}, \vec{z}] = \vec{x} \cdot (\vec{y} \times \vec{z}) \]

Applying this to both sides of the equation, we have:

\[ \lambda(\vec{a} + \vec{b}) \cdot (\lambda^2 \vec{b} \times \lambda \vec{c}) = \vec{a} \cdot ((\vec{b} + \vec{c}) \times \vec{b}) \]

### Step 2: Simplify the Left Side
On the left side, we can simplify:

\[ \lambda(\vec{a} + \vec{b}) \cdot (\lambda^3 (\vec{b} \times \vec{c})) = \lambda^4 (\vec{a} + \vec{b}) \cdot (\vec{b} \times \vec{c}) \]

### Step 3: Simplify the Right Side
Now simplifying the right side:

\[ \vec{a} \cdot ((\vec{b} + \vec{c}) \times \vec{b}) = \vec{a} \cdot (\vec{b} \times \vec{b} + \vec{c} \times \vec{b}) \]

Since \(\vec{b} \times \vec{b} = \vec{0}\), we have:

\[ \vec{a} \cdot (\vec{c} \times \vec{b}) \]

### Step 4: Equate Both Sides
Now we can equate both sides:

\[ \lambda^4 (\vec{a} + \vec{b}) \cdot (\vec{b} \times \vec{c}) = \vec{a} \cdot (\vec{c} \times \vec{b}) \]

### Step 5: Analyze the Equation
We can rewrite the left side:

\[ \lambda^4 (\vec{a} \cdot (\vec{b} \times \vec{c}) + \vec{b} \cdot (\vec{b} \times \vec{c})) \]

Since \(\vec{b} \cdot (\vec{b} \times \vec{c}) = 0\), we have:

\[ \lambda^4 \vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{a} \cdot (\vec{c} \times \vec{b}) \]

### Step 6: Factor Out \(\vec{a}\)
Assuming \(\vec{a}\) is non-zero, we can divide both sides by \(\vec{a}\):

\[ \lambda^4 = 1 \]

### Step 7: Solve for \(\lambda\)
The solutions to this equation are:

\[ \lambda = 1, -1 \]

### Conclusion
Thus, there are exactly two values of \(\lambda\) that satisfy the equation.

### Final Answer
The correct option is: **exactly two values of \(\lambda\)**.

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