Find a unit vector perpendicular to each of the vectors a-b- and a-b- where a-k-b-2-3k-

Given:a⃗=i⃗+j⃗+k⃗ and b⃗=i⃗+2j⃗+ 3k⃗ (a⃗+b⃗)⃗= (î+ĵ+k̂)+(î+2ĵ+3k̂) (a⃗+b⃗)=2î+3ĵ+4k̂ (a⃗ b⃗)=(î+ĵ+k̂) (î+2ĵ+3k̂) a⃗ b⃗=0î ĵ 2k̂ Let c⃗=(a⃗+b⃗)×(a⃗ b⃗) ⇒ ...

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Standard X

History

Course of The First World War - Western War Front

Find a unit v...

Question

Find a unit vector perpendicular to each of the vectors $(\to a + \to b)$ and $(\to a - \to b),$ where $\to a =^i +^j +^k,$
$\to b =^i + 2^j + 3^k$

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Solution

$Given : \to a = \to i + \to j + \to k a n d \to b = \to i + 2 \to j + 3 \to k$

$(\to a + \to b) = (^i +^j +^k) + (^i + 2^j + 3^k)$

$(\to a + \to b) = 2^i + 3^j + 4^k$

$(\to a - \to b) = (^i +^j +^k) - (^i + 2^j + 3^k)$

$\to a - \to b = 0^i -^j - 2^k$

$Let \to c = (\to a + \to b) \times (\to a - \to b)$

$\Rightarrow \to c = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 2 & 3 & 4 0 & - 1 & - 2 \end{matrix} ∣ ∣ ∣ ∣ ∣$

$\to c =^i [(3 \times - 2) - (- 1 \times 4)] -^J [(2 \times - 2) - (0 \times 4)] +^k [(2 \times - 1) - (0 \times 3)]$

$\to c =^i [- 6 - (- 4)] -^j [- 4 - 0] +^k [- 2 - 0]$

$\to c = - 2^i + 4^J - 2^k$

$Unit vector \to c = \frac{\to c}{magnitude of \to c}$

$^c = \frac{(- 2^i + 4^j - 2^k)}{\sqrt{(- 2)^{2} + (4)^{2} + (- 2)^{2}}}$

$^c = \frac{(- 2^i + 4^j - 2^k)}{\sqrt{4 + 16 + 4}}$

$^c = \frac{(c - 2^i + 4^j - 2^k)}{2 \sqrt{6}}$

$Unit vector \to c (^c) = \frac{- 1}{\sqrt{6}}^i + \frac{2}{\sqrt{6}}^j - \frac{1}{\sqrt{6}}^k$

$Hence, the unit vector \to c is \to c = \frac{- 1}{\sqrt{6}}^i + \frac{2}{\sqrt{6}}^j - \frac{1}{\sqrt{6}}^k$

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Similar questions

Q. If

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

and

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

, find a unit vector parallel to the vector

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Q. Find a unit vector perpendicular to each of the vector

\to a + \to b

and

\to a - \to b

, where

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

and

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

Q. Find a unit vector perpendicular to each of the vector

(\to a + \to b)

and

(\to a - \to b)

, where

\to a =^i +^j +^k

and

\to b =^i + 2^j + 3^k .

Q. Find the projection of the vector

\to a

and

\to b

, where

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

and

\to b =^i - 2^j - 3^k

Q. Show that the vectors

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given by

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

,and

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

are non-coplanar.Express vector

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Course of The First World War - Western War Front

Standard X History

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Find a unit vector perpendicular to each of the vectors (→a+→b) and (→a−→b), where →a=^i+^j+^k, →b=^i+2^j+3^k

Find a unit vector perpendicular to each of the vectors $(\to a + \to b)$ and $(\to a - \to b),$ where $\to a =^i +^j +^k,$
$\to b =^i + 2^j + 3^k$