If the vectors `veca, vecb, vecc` are non -coplanar and `l,m,n` are distinct scalars such that
`[(lveca+mvecb+nvecc, lvecb+mvecc+nveca,lvecc+mveca+nvecb)]=0` then

Options

A. `a) l + m + n=0`
B. b) roots of the equation `lx^(2) + mx + n =0` are equal
C. c)`l^(2)+m^(2) + n^(2) =0`
D. d)`l^(3) + m^(2) + n^(3) = 3 lmn `

Explanation

To solve the problem, we need to analyze the given expression involving the vectors \(\vec{a}, \vec{b}, \vec{c}\) and the scalars \(l, m, n\). The expression is given as:

\[ [(l\vec{a} + m\vec{b} + n\vec{c}, l\vec{b} + m\vec{c} + n\vec{a}, l\vec{c} + m\vec{a} + n\vec{b})] = 0 \]

### Step 1: Set up the determinant

We can express the given condition in terms of a determinant of a matrix formed by the coefficients of the vectors. The determinant can be set up as follows:

\[ D = \begin{vmatrix} l & m & n \\ n & l & m \\ m & n & l \end{vmatrix} \]

### Step 2: Calculate the determinant

To calculate the determinant \(D\), we can use the formula for the determinant of a \(3 \times 3\) matrix:

\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Substituting the values from our matrix:

\[ D = l \begin{vmatrix} l & m \\ n & l \end{vmatrix} - m \begin{vmatrix} n & m \\ m & l \end{vmatrix} + n \begin{vmatrix} n & l \\ m & n \end{vmatrix} \]

Calculating each of the \(2 \times 2\) determinants:

1. \(\begin{vmatrix} l & m \\ n & l \end{vmatrix} = l^2 - mn\)
2. \(\begin{vmatrix} n & m \\ m & l \end{vmatrix} = nl - m^2\)
3. \(\begin{vmatrix} n & l \\ m & n \end{vmatrix} = n^2 - ml\)

Thus, we have:

\[ D = l(l^2 - mn) - m(nl - m^2) + n(n^2 - ml) \]

### Step 3: Simplify the determinant

Expanding this gives:

\[ D = l^3 - lmn - mnl + m^3 + n^3 - nml \]

Combining like terms, we get:

\[ D = l^3 + m^3 + n^3 - 3lmn \]

### Step 4: Set the determinant to zero

Since we are given that the determinant equals zero, we have:

\[ l^3 + m^3 + n^3 - 3lmn = 0 \]

### Step 5: Use the identity for sums of cubes

Using the identity for sums of cubes, we can rewrite this as:

\[ (l + m + n)(l^2 + m^2 + n^2 - lm - mn - nl) = 0 \]

### Step 6: Analyze the factors

Since \(l, m, n\) are distinct scalars, \(l + m + n\) cannot be zero. Therefore, we must have:

\[ l^2 + m^2 + n^2 - lm - mn - nl = 0 \]

This implies that the vectors are related in such a way that they form a specific geometric configuration.

### Conclusion

Thus, we conclude that the condition leads to a specific relationship among the scalars \(l, m, n\).