Let - - -2- -k- - - -2- - -3k- and - -2- - -6k- be three vectors- If - and - are both perpendicular to the vector - and - - 10- then what is the magnitude of -

The correct option is A 2√( 3 ) Given #160; α⃗=î+2ĵ k̂ β⃗=2î ĵ+3k̂ γ⃗=2î+ĵ+6k̂ Let #160; δ⃗=aî+bĵ+ck̂ α⃗·δ⃗=0 (î+2ĵ k̂)·(a ...

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Byju's Answer

Standard XII

Mathematics

Properties of Composite Function

Let α =î +2...

Question

Let $\to α =^i + 2^j -^k, \to β = 2^i -^j + 3^k$ and $\to γ = 2^i +^j + 6^k$ be three vectors. If $\to α$ and $\to β$ are both perpendicular to the vector $\to δ$ and $\to δ . \to γ = 10$ , then what is the magnitude of $\to δ$ ?

\sqrt{3}

units

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2 \sqrt{3}

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\frac{\sqrt{3}}{2}

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\frac{1}{\sqrt{3}}

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Solution

The correct option is A $2 \sqrt{3}$
Given
$\to α =^i + 2^j -^k$
$\to β = 2^i -^j + 3^k$
$\to γ = 2^i +^j + 6^k$

Let
$\to δ = a^i + b^j + c^k$

$\to α \cdot \to δ = 0$
$(^i + 2^j -^k) \cdot (a^i + b^j + c^k) = 0$
$a + 2 b - c = 0$ -----(1)

$\to β \cdot \to δ = 0$
$(2^i -^j + 3^k) \cdot (a^i + b^j + c^k) = 0$
$2 a - b + 3 c = 0$ -----(2)
and
$\to γ \cdot \to δ = 10$
$(2^i +^j + 6^k) \cdot (a^i + b^j + c^k) = 10$
$2 a + b + 6 c = 10$ -----(3)

from eq (1)
$a = c - 2 b$
putting a in eq (2)
$2 (c - 2 b) - b + 3 c = 0$
$2 c - 4 b - b + 3 c = 0$
$- 5 b + 5 c = 0$
$- 5 b = - 5 c$
$b = c$
so
$a = c - 2 b$
$a = b - 2 b$
$a = - b$
from eq (3)
$- 2 b + b + 6 b = 10$
$5 b = 10$
$b = 2$

SO
$c = b = 2$
$a = - 2$
$\to δ = - 2^i + 2^j + 2^k$
$∣ ∣ \to δ ∣ ∣ = \sqrt{(- 2)^{2} + 2^{2} + 2^{2}}$
$∣ ∣ \to δ ∣ ∣ = \sqrt{4 + 4 + 4}$
$∣ ∣ \to δ ∣ ∣ = \sqrt{12}$
$∣ ∣ \to δ ∣ ∣ = 2 \sqrt{3}$
Hence option B is correct

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Let →α=^i+2^j−^k,→β=2^i−^j+3^k and →γ=2^i+^j+6^k be three vectors. If →α and →β are both perpendicular to the vector →δ and →δ.→γ=10, then what is the magnitude of →δ?

Let $\to α =^i + 2^j -^k, \to β = 2^i -^j + 3^k$ and $\to γ = 2^i +^j + 6^k$ be three vectors. If $\to α$ and $\to β$ are both perpendicular to the vector $\to δ$ and $\to δ . \to γ = 10$ , then what is the magnitude of $\to δ$ ?