`veca , vec b , vec c ` are non-coplanar vectors and ` x vec a + y vec b + z vec c = vec 0` then

Options

A. at least of one of x, y, z is zero
B. x, y, z are necessarily zero
C. none of them are zero
D. none of these

Answer

To solve the problem, we need to analyze the equation given:

\[ x \vec{a} + y \vec{b} + z \vec{c} = \vec{0} \]

where \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar vectors. We want to find the conditions under which this equation holds true.

### Step 1: Understanding Non-Coplanar Vectors
Non-coplanar vectors are vectors that do not lie in the same plane. This means that they are linearly independent. For vectors to be non-coplanar, the only solution to the linear combination of these vectors equaling zero must be the trivial solution.

### Step 2: Setting Up the Equation
Given the equation:

\[ x \vec{a} + y \vec{b} + z \vec{c} = \vec{0} \]

we can interpret this as a linear combination of the vectors \( \vec{a}, \vec{b}, \vec{c} \).

### Step 3: Analyzing the Linear Combination
Since \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar, the only way for their linear combination to equal the zero vector is if the coefficients \( x, y, z \) are all zero. This is a property of linearly independent vectors.

### Step 4: Conclusion
Thus, we can conclude that:

\[ x = 0, \quad y = 0, \quad z = 0 \]

This means that for the equation \( x \vec{a} + y \vec{b} + z \vec{c} = \vec{0} \) to hold true, the coefficients \( x, y, z \) must all be zero.

### Final Answer
The correct conclusion is that \( x = 0, y = 0, z = 0 \) when \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar vectors.

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