Find the projection of 3- - 4- - k- on - - - - k- and its magnitude-

Let a⃗ = lt; 3,4,1 gt; and b⃗ = lt; 1,1, 1 gt; then vector projection of a⃗ onto b⃗ using the following formula: proj _ b ...

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Standard XII

Mathematics

Applications of Dot Product

Find the proj...

Question

Find the projection of $3^i + 4^j +^k$ on $^i +^j -^k$ and its magnitude.

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Solution

Let $\to a =< 3, 4, 1 >$ and $\to b =< 1, 1, - 1 >$
then vector projection of $\to a$ onto $\to b$ using the following formula:
${proj}_{\to b} \to a = (\frac{\to a \cdot \to b}{| \to b |}) \frac{\to b}{| \to b |}$
${proj}_{\to b} \to a = \frac{6}{\sqrt{3}} \cdot \frac{< 1, 1, - 1 >}{\sqrt{3}}$
$\Rightarrow< 2, 2, - 2 >$
Magnitude is $2 \sqrt{3}$

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Find the projection of 3^i+4^j+^k on ^i+^j−^k and its magnitude.

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Find the projection of $3^i + 4^j +^k$ on $^i +^j -^k$ and its magnitude.