Let a- - 3- - 2- - xk- and b- - - - - - k- for some real X- Then -a-b- - r is possible if-

The correct option is C r ≥ 5 √(32) a⃗×b⃗ = #160;î amp;ĵ amp;k̂ 3 amp;2 amp;x 1 amp; 1 amp;1 = (2 +x)î + (x 3)ĵ 5k ...

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Byju's Answer

Standard XII

Mathematics

Determinant

Let a⃗ = 3î...

Question

Let $\to a = 3^i + 2^j + x^k$ and $\to b =^i -^j +^k$ , for some real $X$ . Then $| \to a \times \to b | = r$ is possible if:

3 \sqrt{\frac{3}{2}} < r < 5 \sqrt{\frac{3}{2}}

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0 < r \leq \sqrt{\frac{3}{2}}

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\sqrt{\frac{3}{2}} < r \leq 3 \sqrt{\frac{3}{2}}

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r \geq 5 \sqrt{\frac{3}{2}}

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Solution

The correct option is C $r \geq 5 \sqrt{\frac{3}{2}}$
$\to a \times \to b = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 3 & 2 & x 1 & - 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= (2 + x)^i + (x - 3)^j - 5 k$
$| \to a \times \to b | = \sqrt{4 v + x^{2} + 4 x + x^{2} + 9 - 6 x + 25}$
$= \sqrt{2 x^{2} - 2 x + 38}$
$\Rightarrow | \to a \times \to b | \geq \sqrt{\frac{75}{2}}$
$\Rightarrow | \to a \times \to b | \geq 5 \sqrt{\frac{3}{2}}$

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Similar questions

Q. Let

\to a = 3^i + 2^j + x^k

and

\to b =^i -^j +^k

, for some real

x

. Then

| \to a \times \to b | = r

is possible if :

Q.
I : If the vectors

\to a = (1, x, - 2)

\to b = (x, 3, - 4)

are mutually perpendicular, then

x = 2

II : If

\to a =^i + 2^j + 3^k, \to b = -^i + 2^j +^k, \to c = 3^i +^j

and

\to a + t \to b

is perpendicular to

\to c

then

t = 5

Q. Let

\to a

= 2

^i

^j

^k

\to b

^i

+ 2

^j

^k

and

\to c

^i

^j

^k

be three vector . A vector in plane of

\to b

and

\to c

, whose projection on

\to a

is a magnitude

\sqrt{2 / 3}

, is

Q. If

\to c = 2 \to a - 3 \to b

where

| \to a | = 2

and

| \to b | = 3

and

\to a \cdot \to b = 6

, also

\to c

is coplanar with

\to p =^i +^j - 2^k

and

\to q =^i - 2^j +^k

and perpendicular to

\to r =^i +^j + 2^k

then

\to c

can be

Q. Let

\to a =^j -^k

and

\to c =^i -^j -^k

. Then vector

\to b

satisfying

\to a \times \to b + \to c = \to 0

and

\to a \cdot \to b = 3

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Let →a=3^i+2^j+x^k and →b=^i−^j+^k, for some real X. Then |→a×→b|=r is possible if:

The correct option is C r≥5√32→a×→b=∣∣ ∣ ∣∣^i^j^k32x1−11∣∣ ∣ ∣∣=(2+x)^i+(x−3)^j−5k|→a×→b|=√4v+x2+4x+x2+9−6x+25=√2x2−2x+38⇒|→a×→b|≥√752⇒|→a×→b|≥5√32

Let $\to a = 3^i + 2^j + x^k$ and $\to b =^i -^j +^k$ , for some real $X$ . Then $| \to a \times \to b | = r$ is possible if: