The set of values of m for which vectors 2x-x-k- and -x-m-2k- makes obtuse angle for all values of x- is

Given : a=2xî xĵ k̂ and b= xî+mĵ+2k̂ For obtuse angle between a and b: a·b lt;0 ⇒ (2xî xĵ k̂)·( xî+mĵ+2k̂) lt;0 ⇒ 2x^2 mx 2 lt;0; ∀ x∈ℝ ⇒ D lt;0 [∵ cof ...

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Standard X

Mathematics

Section Formula

The set of va...

Question

The set of value(s) of $m$ for which vectors $2 x^i - x^j -^k$ and $- x^i + m^j + 2^k$ makes obtuse angle for all values of $x \in R,$ is

(- 6, 4)

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(4, \infty)

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(- 4, 4)

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(- 5, 5)

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Solution

The correct option is C $(- 4, 4)$
Given :
$\to a = 2 x^i - x^j -^k$ and $\to b = - x^i + m^j + 2^k$
For obtuse angle between $\to a$ and $\to b : \to a \cdot \to b < 0$
$\Rightarrow (2 x^i - x^j -^k) \cdot (- x^i + m^j + 2^k) < 0 \Rightarrow - 2 x^{2} - m x - 2 < 0; \forall x \in R \Rightarrow D < 0 [∵ coff. x^{2} < 0] \Rightarrow m^{2} - 16 < 0 \Rightarrow m \in (- 4, 4)$

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Section Formula

Standard X Mathematics

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The set of value(s) of m for which vectors 2x^i−x^j−^k and −x^i+m^j+2^k makes obtuse angle for all values of x∈R, is

The correct option is C (−4,4)Given : →a=2x^i−x^j−^k and →b=−x^i+m^j+2^k For obtuse angle between →a and →b:→a⋅→b<0 ⇒(2x^i−x^j−^k)⋅(−x^i+m^j+2^k)<0⇒−2x2−mx−2<0;∀x∈R⇒D<0 [∵ coff. x2<0]⇒m2−16<0⇒m∈(−4,4)

The set of value(s) of $m$ for which vectors $2 x^i - x^j -^k$ and $- x^i + m^j + 2^k$ makes obtuse angle for all values of $x \in R,$ is