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Hi, I came across this statement while I was reading an article on resolution of vector in arbitrary basis.
"v = αa + βb + γc ---(1) , where a, b and c are three independent vectors.
we observe that the coefficient α cannot involve any overlap of v with either b or c ; β cannot
involve any overlap of v with either c or a ; and γ cannot involve any overlap of v with either
1a or b. Therefore α must be proportional to that part of v which lies along ( b × c ), i.e., to
[ (b × c) · v ]. Similar conclusions hold good for β and γ."
This is a part of the proof that says v = [a [ (b × c) · v ] + b [ (c × a) · v ] + c [ (a × b) · v ]]/[ (a × b) · c ]
My question is, how can we ascertain that " the coefficient α cannot involve any overlap of v with either b or c" from the given equation (1)
Here is the link to what I was reading in page 2,
http://www.ias.ac.in/resonance/Volumes/01/10/0006-0013.pdf
"v = αa + βb + γc ---(1) , where a, b and c are three independent vectors.
we observe that the coefficient α cannot involve any overlap of v with either b or c ; β cannot
involve any overlap of v with either c or a ; and γ cannot involve any overlap of v with either
1a or b. Therefore α must be proportional to that part of v which lies along ( b × c ), i.e., to
[ (b × c) · v ]. Similar conclusions hold good for β and γ."
This is a part of the proof that says v = [a [ (b × c) · v ] + b [ (c × a) · v ] + c [ (a × b) · v ]]/[ (a × b) · c ]
My question is, how can we ascertain that " the coefficient α cannot involve any overlap of v with either b or c" from the given equation (1)
Here is the link to what I was reading in page 2,
http://www.ias.ac.in/resonance/Volumes/01/10/0006-0013.pdf
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