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Test for Coplanarity

A vector d⃗ i...

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. If $\to r = 3 \to x + 4 \to y + 5 \to z,$ then the value of $\to d . \to r$ is

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Solution

$\to a =^i -^j +^k$ ,
$\to b = 2^i +^j$
and $\to c = 3^j - 2^k$
Since $[\to a \to b \to c] = ∣ ∣ ∣ ∣ \begin{matrix} 1 & - 1 & 1 2 & 1 & 0 0 & 3 & - 2 \end{matrix} ∣ ∣ ∣ ∣ = 0$
$∴ \to a, \to b$ and $\to c$ are coplanar vectors.
Further, since $\to d$ is equally inclined to $\to a, \to b$ and $\to c$ , we have
$∴ \to d . \to a = \to d . \to b = \to d . \to c = 0$
$∴ \to d . \to x = \to d . \to y = \to d . \to z = 0$
$∴ \to d . \to r = 0$

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Test for Coplanarity

Standard XII Mathematics

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