If a-k- b-2-2k-Find a unit vector c orthogonal to a and b

The correct option is C k̂ ĵ√( 2 ) Since a⃗, b⃗ and c⃗ are three vectorsGiven a⃗=î+ĵ+k̂ , b⃗=2ĵ+2k̂ |a⃗×b⃗|=î amp; ĵ amp; ...

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Standard XII

Mathematics

Continuity of a Function

If a=î+ĵ+k̂...

Question

If $\to a =^i +^j +^k, \to b = 2^j + 2^k$ .Find a unit vector $\to c$ orthogonal to $\to a$ and $\to b$

\frac{^i +^j}{\sqrt{2}}

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\frac{^j +^k}{\sqrt{2}}

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\frac{^i -^k}{\sqrt{2}}

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\frac{^k -^j}{\sqrt{2}}

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Solution

The correct option is C $\frac{^k -^j}{\sqrt{2}}$
Since $\to a, \to b$ and $\to c$ are three vectors

Given $\to a =^i +^j +^k$ , $\to b = 2^j + 2^k$
$| \to a \times \to b | = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 1 & 1 & 1 0 & 2 & 2 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$=^i (2 - 2) -^j (2 - 0) +^k (2 - 0)$
$= - 2^j + 2^k$
$= (0, - 2, 2)$

$| \to a \times \to b | = \sqrt{0 + 4 + 4} = 2 \sqrt{2}$

Unit vector $= \frac{1}{2 \sqrt{2}} (0, - 2, 2)$

$= (0, \frac{- 1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$

Hence $\to c = \frac{- j + k}{\sqrt{2}}$ .

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If →a=^i+^j+^k,→b=2^j+2^k.Find a unit vector →c orthogonal to →a and →b

If $\to a =^i +^j +^k, \to b = 2^j + 2^k$ .Find a unit vector $\to c$ orthogonal to $\to a$ and $\to b$