Find the projection of a-2b on c where a-2-2-k- b-2-2k- and c-2-4k-

Given, nbsp; a=2î 2ĵ+k̂, b=î+2ĵ 2k̂ and c=2î ĵ+4k̂ We need to find nbsp;the projection of (a+2b) on c We know that the projection of (a+2b) on c is given by ...

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Standard XII

Physics

Multiplication with Vectors

Find the proj...

Question

Find the projection of $(\to a + 2 \to b)$ on $\to c$ where $\to a = 2^i - 2^j +^k, \to b =^i + 2^j - 2^k$ and $\to c = 2^i -^j + 4^k$

\frac{8}{\sqrt{21}}

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\frac{- 8}{\sqrt{21}}

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\frac{6}{\sqrt{21}}

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\frac{- 6}{\sqrt{21}}

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Solution

The correct option is D $\frac{- 6}{\sqrt{21}}$
Given, $\to a = 2^i - 2^j +^k, \to b =^i + 2^j - 2^k$ and $\to c = 2^i -^j + 4^k$
We need to find the projection of $(\to a + 2 \to b)$ on $\to c$
We know that the projection of $(\to a + 2 \to b)$ on $\to c$ is given by $(\to a + 2 \to b) .^c$ , where $^c$ is the unit vector of $\to c$ .
So, we have
$^c = \frac{\to c}{| \to c |} = \frac{2^i -^j + 4^k}{\sqrt{2^{2} + (- 1)^{2} + 4^{2}}} = \frac{1}{\sqrt{21}} (2^i -^j + 4^k)$
Also, $(\to a + 2 \to b) = (2^i - 2^j +^k) + 2 (^i + 2^j - 2^k)$
$\Rightarrow (\to a + 2 \to b) = 4^i + 2^j - 3^k$
Hence, projection is given by
$(\to a + 2 \to b) .^c = (4^i + 2^j - 3^k) . (\frac{1}{\sqrt{21}} (2^i -^j + 4^k))$
$\Rightarrow \frac{8 - 2 - 12}{\sqrt{21}} = \frac{- 6}{\sqrt{21}}$

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Find the projection of (→a+2→b) on →c where →a=2^i−2^j+^k,→b=^i+2^j−2^k and →c=2^i−^j+4^k

Find the projection of $(\to a + 2 \to b)$ on $\to c$ where $\to a = 2^i - 2^j +^k, \to b =^i + 2^j - 2^k$ and $\to c = 2^i -^j + 4^k$