Let a,b,c be any threee non zero non-coplanar vectors, then any vector `r` is equal to where
`x=[[[vecr,vecb, vecc]]]/[[[veca ,vecb, vecc]]], y=[[[vecr ,vecc, veca]]]/[[[veca, vecb ,vecc]]], z=[[[vecr, veca ,vecb]]]/[[[veca, vecb, vecc]]]`

Options

A. `za + xb + yc`
B. `xa + yb + zc`
C. `ya + zb + xc`
D. none of these

Answer

To solve the problem, we need to express the vector \( \mathbf{r} \) in terms of the non-coplanar vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \). The given equations for \( x, y, z \) are:

\[ x = \frac{[\mathbf{r}, \mathbf{b}, \mathbf{c}]}{[\mathbf{a}, \mathbf{b}, \mathbf{c}]}, \quad y = \frac{[\mathbf{r}, \mathbf{c}, \mathbf{a}]}{[\mathbf{a}, \mathbf{b}, \mathbf{c}]}, \quad z = \frac{[\mathbf{r}, \mathbf{a}, \mathbf{b}]}{[\mathbf{a}, \mathbf{b}, \mathbf{c}]} \]

Where \( [\mathbf{u}, \mathbf{v}, \mathbf{w}] \) denotes the scalar triple product of the vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w} \).

### Step-by-Step Solution:

1. **Assume a Linear Combination**:
Since \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are non-coplanar, we can express \( \mathbf{r} \) as a linear combination of these vectors:
\[ \mathbf{r} = r_1 \mathbf{a} + r_2 \mathbf{b} + r_3 \mathbf{c} \]

2. **Dot Product with \( \mathbf{b} \times \mathbf{c} \)**:
Taking the dot product of \( \mathbf{r} \) with \( \mathbf{b} \times \mathbf{c} \):
\[ \mathbf{r} \cdot (\mathbf{b} \times \mathbf{c}) = (r_1 \mathbf{a} + r_2 \mathbf{b} + r_3 \mathbf{c}) \cdot (\mathbf{b} \times \mathbf{c}) \]
Using properties of the dot product, we know:
\[ \mathbf{b} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \quad \text{and} \quad \mathbf{c} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \]
Thus, we have:
\[ \mathbf{r} \cdot (\mathbf{b} \times \mathbf{c}) = r_1 (\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})) = r_1 [\mathbf{a}, \mathbf{b}, \mathbf{c}] \]

3. **Express \( r_1 \)**:
From the above, we can express \( r_1 \):
\[ r_1 = \frac{\mathbf{r} \cdot (\mathbf{b} \times \mathbf{c})}{[\mathbf{a}, \mathbf{b}, \mathbf{c}]} = x \]

4. **Repeat for \( r_2 \) and \( r_3 \)**:
Similarly, we can find \( r_2 \) and \( r_3 \) by taking the dot products with \( \mathbf{c} \times \mathbf{a} \) and \( \mathbf{a} \times \mathbf{b} \) respectively:
\[ r_2 = \frac{\mathbf{r} \cdot (\mathbf{c} \times \mathbf{a})}{[\mathbf{a}, \mathbf{b}, \mathbf{c}]} = y \]
\[ r_3 = \frac{\mathbf{r} \cdot (\mathbf{a} \times \mathbf{b})}{[\mathbf{a}, \mathbf{b}, \mathbf{c}]} = z \]

5. **Final Expression for \( \mathbf{r} \)**:
Therefore, we can express \( \mathbf{r} \) as:
\[ \mathbf{r} = x \mathbf{a} + y \mathbf{b} + z \mathbf{c} \]

### Summary:
The vector \( \mathbf{r} \) can be expressed as a linear combination of the non-coplanar vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) using the coefficients \( x, y, z \) derived from the scalar triple products.