Find the projection vector of 2 -k- on -2 -k-

As projection vector of a⃗ on b⃗ = (a⃗. b̂)b̂ So, projection of 2 î ĵ + k̂ on î 2 ĵ + k̂ is [(2 î ĵ + k̂). (î 2 ĵ + k̂/√(1^2 + ( 2)^2 + 1^2))] (î 2 ĵ ...

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Byju's Answer

Standard XII

Mathematics

Properties of Conjugate of a Complex Number

Find the proj...

Question

Find the projection (vector) of $2^i -^j +^k$ on $^i - 2^j +^k .$

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Solution

As projection vector of $\to a$ on $\to b = (\to a .^b)^b$
So, projection of $2^i -^j +^k$ on $^i - 2^j +^k$ is $[(2^i -^j +^k) . (\frac{^i - 2^j +^k}{\sqrt{1^{2} + (- 2)^{2} + 1^{2}}})] (\frac{^i - 2^j +^k}{\sqrt{1^{2} + (- 2)^{2} + 1^{2}}})$
i.e., $= [(2^i -^j +^k) . (\frac{^i - 2^j +^k}{\sqrt{6}})] (\frac{^i - 2^j +^k}{\sqrt{6}}) = (\frac{2 + 2 + 1}{\sqrt{6}}) (\frac{^i - 2^j +^k}{\sqrt{6}}) = \frac{5 (^i - 2^j +^k)}{6}) .$

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Find the projection (vector) of 2^i−^j+^k on ^i−2^j+^k.

As projection vector of →a on →b=(→a.^b)^b So, projection of 2^i−^j+^k on ^i−2^j+^k is [(2^i−^j+^k).(^i−2^j+^k√12+(−2)2+12)](^i−2^j+^k√12+(−2)2+12) i.e., =[(2^i−^j+^k).(^i−2^j+^k√6)](^i−2^j+^k√6)=(2+2+1√6)(^i−2^j+^k√6)=5(^i−2^j+^k)6).

Find the projection (vector) of $2^i -^j +^k$ on $^i - 2^j +^k .$