Leta - 2- - - k- b - - - 2- k-and a unit vector c be coplanar- If c is perpendicular to a- then c is equal to-

C coplanar with a and b So, c = xa+ yb c = x(2î + ĵ+ k̂) + y( î + 2ĵ k̂) c = (2x+y)î +(x+2y) ĵ+(x y) k̂ And, c.a = 0 ( (2x+y)î +(x+2y) ĵ+(x y) k̂). (2î + ĵ+ k̂ ...

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Standard XII

Mathematics

Equation of a Plane Passing through Three Points

Leta = 2î + ĵ...

Question

Let $\to a = 2^i +^j +^k, \to b =^i + 2^j -^k$ and a unit vector $\to c$ be coplanar. If $\to c$ is perpendicular to $\to a$ , then $\to c$ is equal to:

\pm \frac{1}{3 \sqrt{2}} (-^j +^k)

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\pm \frac{1}{\sqrt{2}} (-^j +^k)

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\pm \frac{3}{\sqrt{2}} (-^j +^k)

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\pm \frac{1}{2 \sqrt{2}} (-^j +^k)

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Solution

The correct option is B $\pm \frac{1}{\sqrt{2}} (-^j +^k)$
$\to c$ coplanar with $\to a$ and $\to b$
So, $\to c = x \to a + y \to b$
$\to c = x (2^i +^j +^k) + y (^i + 2^j -^k)$
$\to c = (2 x + y)^i + (x + 2 y)^j + (x - y)^k$
And, $\to c . \to a = 0$
$((2 x + y)^i + (x + 2 y)^j + (x - y)^k) . (2^i +^j +^k) = 0$
$2 (2 x + y) + (x + 2 y) + (x - y) = 0$
$4 x + 2 y + x + 2 y + x - y = 0$
$6 x + 3 y = 0$
$y = - 2 x$
So, $\to c = 0^i - 3 x^j + 3 x^k$
And we know, $| \to c | = 1$
$\sqrt{0 + (- 3 x)^{2} + (3 x)^{2}} = 1$
$18 x^{2} = 1$
$x = \pm \frac{1}{3 \sqrt{2}}$
$\to c = 3 x (-^j +^k) = 3 (\pm \frac{1}{3 \sqrt{2}}) (-^j +^k)$
$∴ \to c = \pm \frac{1}{\sqrt{2}} (-^j +^k)$

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Equation of a Plane Passing through Three Points

Standard XII Mathematics

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Let →a=2^i+^j+^k, →b=^i+2^j−^k and a unit vector →c be coplanar. If →c is perpendicular to →a, then →c is equal to:

Let $\to a = 2^i +^j +^k, \to b =^i + 2^j -^k$ and a unit vector $\to c$ be coplanar. If $\to c$ is perpendicular to $\to a$ , then $\to c$ is equal to: