Let a→=i^−k^,b→=xi^+j^+(1−x)k^ and c→=yi^+xj^+(1+x−y)k^ then a→,b→ and c→ are non-coplanar for

a→=i^−k^ and b→=xi^+j^+(1−x)k^c→=yi^+xj^+(1+x−y)k^[a→ b→ c→]=[10−1x11−xyx1+x−y]=(1+x−y−x+x2)−1(x2−y2)=1

Let a→=i^−k^,b→=xi^+j^+(1−x)k^ and c→=yi^+xj^+(1+x−y)k^ then a→,b→ and c→ are non-coplanar for

a→=i^−k^ and b→=xi^+j^+(1−x)k^c→=yi^+xj^+(1+x−y)k^[a→ b→ c→]=[10−1x11−xyx1+x−y]=(1+x−y−x+x2)−1(x2−y2)=1

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Let $\vec{a} = \hat{i} - \hat{k}, \vec{b} = x \hat{i} + \hat{j} + (1 - x) \hat{k}$ and $\vec{c} = y \hat{i} + x \hat{j} + (1 + x - y) \hat{k}$ then $\vec{a}, \vec{b}$ and $\vec{c}$ are non-coplanar for

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some value of $x$

some values of $y$

no values of $x$ and $y$

for all values of $x$ and $y$

answer is D.

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$\vec{a} = \hat{i} - \hat{k}$ and $\vec{b} = x \hat{i} + \hat{j} + (1 - x) \hat{k}$

$\vec{c} = y \hat{i} + x \hat{j} + (1 + x - y) \hat{k}$

$[\vec{a} \vec{b} \vec{c}] = [\begin{matrix} 1 & 0 & - 1 \\ x & 1 & 1 - x \\ y & x & 1 + x - y \end{matrix}] = (1 + x - y - x + x^{2}) - 1 (x^{2} - y^{2}) = 1$

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Let a→=i^−k^,b→=xi^+j^+(1−x)k^ and c→=yi^+xj^+(1+x−y)k^ then a→,b→ and c→ are non-coplanar for

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Let $\vec{a} = \hat{i} - \hat{k}, \vec{b} = x \hat{i} + \hat{j} + (1 - x) \hat{k}$ and $\vec{c} = y \hat{i} + x \hat{j} + (1 + x - y) \hat{k}$ then $\vec{a}, \vec{b}$ and $\vec{c}$ are non-coplanar for