Question

Resolution of vectors:
Let A, B, C be non-coplanar 'unit' vectors and a, b, c be scalars such that -
rR = aA + bB + cC
where r is a scalar
and R is a 'unit' vector

Given that R makes an angle of α, β, γ with A, B, C respectively ; find the value of a, b, c in terms of r.

Answer 1

(cos α )^2+(cos β )^2+(cos γ )^2=1

cos α=rR.A/a
somthing like this

Answer 2

(cos α )^2+(cos β )^2+(cos γ )^2=1

I think this equation is valid only for rectangular co-ordinates (and I am working with skew)

Answer 3

forget the equation
use angle of direction law

Answer 4

I have solved the same for 2D system


r sin(α) = b sin(α+β)
Similarly,
r sin(β) = a sin(α+β)

But I can extend the same to 3D system :(

Answer 5

*can't

Answer 6

BTW "what's angle of direction law" ??

Answer 7

u knw angle btw i ,j,k
r called angle of direction

Answer 8

btw any vector+i,j,k

Answer 9

for a vector v=<v1,v2,v3>
direction angels α, β, γ
cos α=v./||v|| ||i||
and so on

Answer 10

v.i sry not v alone

Answer 11

cos α=v.i/||v|| ||i||
cos β =v.j/||v|| ||j||
cos γ=v.k/||v|| ||k||

Answer 12

But I don't have the value of v.i ; v.j ; v.k

Answer 13

:|

Answer 14

lol its only a hint

Answer 15

A,B,C unit vector use them insted of i,j,k
rR is ur v

Answer 16

There's a small problem;
for rectangular co-ordinate system ; we can easily say v.i = v1

But in this case;
v.i = r a cos(α)
v.i = a + b(B.A) + c(C.A)

Now I don't know the angle b/w B and A so I cant solve (B.A) ; same goes for (C.A)

Answer 17

no it not a problem cuz in ur qs the definition of α, β, γ
is for ABC not i j k

Answer 18

i guess it will be like this
cos α=a/r
cos β =b/r
cos γ=c/r

Answer 19

Oh..I should have written it like this
rR.A = r a cos(α)
rR.A = a + b(B.A) + c(C.A)

Now can you see where am I stuck ?

Answer 20

Sorry,
rR.A = r cos(α)
rR.A = a + b(B.A) + c(C.A)

So cos(α) = ( a + b(B.A) + c(C.A) ) / r

Answer 21

:| idk im confused with this too
ill try it ltr

Answer 22

Thanks for your time :)

Answer 23

np lol , its v intresting qs :o

Answer 24

try this book for vecor calcules
Calculus.Early.Transcendentals.10th.edition.by.Stephen.Davis.Howard.Anton.and.Irl.Bivens
very helpfull

Answer 25

Thanxx ^_^

Answer 26

Just to clarify...the problem is not yet solved XD

Answer 27

The formula will finally lead to
cos(α) = ( a + b(B.A) + c(C.A) ) / r

in which IDK the value of (B.A) and (C.A)

Answer 28

Projection of R on B may not be equal to b

We can resolve aA into components along Cartesian co-ordinates but (I think) the final answer should be independent of the choice of coordinate system.

Answer 29

It's not a textbook Q..I just wanted to find the general formula for resolving a 3D vector along any three non-coplanar vectors.

Answer 30

Notation
Small letters denote magnitude
Capital letters denote Unit Vectors

We have
r R = a A + b B + c C

We are given the value of r and the angles R makes with A, B, C.

I need to find the value of a, b, c

I have already solved this for 2-dimension (see 4th post from top) ; but can't extend the solution to 3-dimension.

Answer 31

In your notations ;
I need to find the magnitude of vector a in terms of magnitude of resultant vector.