A vector d- is equally inclined to three vectors a -k- b -2 - and c -3 -2 k- Let x - y - z be three vectors in the plane ofa - b - b - c - c - a- respectively- If r -3 x -4 y -5 z- then the value of d - r is

A =î ĵ+k̂, b=2 î+ĵ and c =3ĵ 2 k̂ Since [abc] = 1 amp; 1 amp; 1 nbsp; 2 amp; 1 amp; 0 nbsp; 0 amp; 3 amp; 2 =0 ∴a, b and c are coplan ...

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Standard XII

Mathematics

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