A vector d is equally inclined to three vectors a - - - - - k- b - 2- - - and c - 3-2k- Let x- y- z be three vectors in the plane of a- b- b- c- c- a- respectively- Then which of the following is INCORRECT-

A =î ĵ+k̂, b=2 î+ĵ c =3ĵ 2 k̂ As, [ab nbsp; c] = 1 amp; 1 amp; 1 nbsp; 2 amp; 1 amp; 0 nbsp; 0 amp; 3 amp; 2 =0 Therefore, a, b and ...

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Byju's Answer

Standard XI

Mathematics

Test for Coplanarity

A vector d is...

Question

A vector $(\to d)$ is equally inclined to three vectors $\to a =^i -^j +^k, \to b = 2^i +^j$ and $\to c = 3^j - 2^k$ . Let $\to x, \to y, \to z$ be three vectors in the plane of $\to a, \to b; \to b, \to c; \to c, \to a$ , respectively. Then which of the following is INCORRECT?

\to x . \to d = 0

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\to y . \to d = 0

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\to z . \to d = 0

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none of these

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Solution

The correct option is D none of these
$\to a =^i -^j +^k$ ,
$\to b = 2^i +^j$
$\to c = 3^j - 2^k$
As,
$[\to a \to b \to c] = ∣ ∣ ∣ ∣ \begin{matrix} 1 & - 1 & 1 2 & 1 & 0 0 & 3 & - 2 \end{matrix} ∣ ∣ ∣ ∣ = 0$
Therefore, $\to a, \to b$ and $\to c$ are coplanar vectors.

As $\to d$ is equally inclined to $\to a, \to b$ and $\to c$ , so $\to d$ should be normal to the plane containing vector $\to a, \to b, \to c$

$∴ \to d . \to a = \to d . \to b = \to d . \to c = 0$
$∴ \to d . \to x = \to d . \to y = \to d . \to z = 0$

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A vector (→d) is equally inclined to three vectors →a=^i−^j+^k, →b=2^i+^j and →c=3^j−2^k. Let →x,→y,→z be three vectors in the plane of →a,→b;→b,→c;→c,→a, respectively. Then which of the following is INCORRECT?

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