Let r- - a-b- sin x - b-c-cos y - 2 c-a- where a-b-c- are three noncoplanar vectors- If r- is perpendicular to a- - b- - c- then minimum value of x2 - y2 is

The correct option is A 13π^24 Given #160; r⃗=(a⃗×b⃗)sin x+(b⃗×c⃗)cos y+2(c⃗×a⃗) — (1) r⃗ is perpendicular to a⃗+b⃗+c⃗ SO #160; r⃗·( ...

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Byju's Answer

Standard XII

Mathematics

Square Root of a Complex Number

Let r⃗ = a⃗...

Question

Let $\to r = (\to a \times \to b) sin x + (\to b \times \to c) cos y + 2 (\to c \times \to a)$ where $\to a \to b \to c$ are three noncoplanar vectors. If $\to r$ is perpendicular to $\to a + \to b + \to c$ , then minimum value of $x^{2} + y^{2}$ is

\frac{13 π^{2}}{4}

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\frac{π^{2}}{4}

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\frac{5 π^{2}}{4}

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None of these

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Solution

The correct option is A $\frac{13 π^{2}}{4}$
Given
$\to r = (\to a \times \to b) sin x + (\to b \times \to c) cos y + 2 (\to c \times \to a)$ ----(1)
$\to r$ is perpendicular to $\to a + \to b + \to c$
SO
$\to r \cdot (\to a + \to b + \to c) = 0$
$((\to a \times \to b) sin x + (\to b \times \to c) cos y + 2 (\to c \times \to a)) \cdot (\to a + \to b + \to c) = 0$

$((\to a \times \to b) \cdot \to a sin x + (\to b \times \to c) \cdot \to a cos y + 2 (\to c \times \to a) \cdot \to a) + ((\to a \times \to b) \cdot \to b sin x + (\to b \times \to c) \cdot \to b cos y + 2 (\to c \times \to a) \cdot \to b) +$
$((\to a \times \to b) \cdot \to c sin x + (\to b \times \to c) \cdot \to c cos y + 2 (\to c \times \to a) \cdot \to c) = 0$

$[\to a \to b \to a] sin x + [\to b \to c \to a] cos y + 2 [\to c \to a \to a] + [\to a \to b \to b] sin x + [\to b \to c \to b] cos y + 2 [\to c \to a \to b] + [\to a \to b \to c] sin x +$
$[\to b \to c \to c] cos y + 2 [\to c \to a \to c] = 0$

but $[\to a \to b \to a] = [\to b \to c \to a] = [\to b \to c \to c] = [\to b \to c \to b] == [\to a \to b \to b] = [\to a \to c \to c] = 0$
$0 + [\to a \to b \to c] cos y + 0 + 0 + 0 + 2 [\to a \to b \to c] + [\to a \to b \to c] sin x + 0 + 0 = 0$
$[\to a \to b \to c] cos y + 2 [\to a \to b \to c] + [\to a \to b \to c] sin x = 0$
$[\to a \to b \to c] (cos y + 2 + sin x) = 0$
$sin x + cos y = - 2$
possible value of
$sin x = - 1$
$x = \frac{3 π}{2}$
$cos y = - 1$
$y = π$
$x^{2} + y^{2} = \frac{13 π^{2}}{4}$

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Similar questions

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Let →r=(→a×→b)sinx+(→b×→c)cosy+2(→c×→a) where →a→b→c are three noncoplanar vectors. If →r is perpendicular to →a+→b+→c, then minimum value of x2+y2 is

Let $\to r = (\to a \times \to b) sin x + (\to b \times \to c) cos y + 2 (\to c \times \to a)$ where $\to a \to b \to c$ are three noncoplanar vectors. If $\to r$ is perpendicular to $\to a + \to b + \to c$ , then minimum value of $x^{2} + y^{2}$ is