Let a- - i- - k- b- - xi- - j- - 1 - x k- and c- - y i- - xj- - 1 - x - y k- then a- b- and c- are non-coplanar for

The correct option is D For all values of x and y a⃗ = i k b⃗ = x #160;i + #160;j + (1 x) #160;k c⃗ #160;= yi⃗ + x j⃗ + ( ...

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Byju's Answer

Standard XII

Mathematics

Condition for Coplanarity of Four Points

Let a⃗ = i⃗...

Question

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

Some values of

x

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Some values of

y

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No values of

x

and

y

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For all values of

x

and

y

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Solution

The correct option is D For all values of $x$ and $y$
$\to a = ˆ i - ˆ k$
$\to b = x ˆ i + ˆ j + (1 - x) ˆ k$
$\to c = y \to i + x \to j + (1 + x - y) \to k$
Consider,
$∣ ∣ ∣ ∣ \begin{matrix} 1 & 0 & - 1 x & 1 & 1 - x y & x & 1 + x - y \end{matrix} ∣ ∣ ∣ ∣$

$= 1 + x - y - x^{2} + y - x + x^{2}$
$= 1$
$\neq 0$ , hence non coplanar for all values of $x$ and $y$ .

Suggest Corrections

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

The correct option is D For all values of x and y→a=ˆi−ˆk→b=xˆi+ˆj+(1−x)ˆk→c=y→i+x→j+(1+x−y)→kConsider,∣∣ ∣∣10−1x11−xyx1+x−y∣∣ ∣∣=1+x−y−x2+y−x+x2=1≠0, hence non coplanar for all values of x and y.

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