Help Needed Solving For Two Scalars Such That au + bw = v

[Jack Bateman]

Homework Statement

Find a and b such that v= au + bw given that v = <1, 1>

Homework Equations

v = au + bw, v = <1,1>

The Attempt at a Solution

I have tried to solve for the appropriate values via guess and check, but it hasn't worked out. I cannot find a proportion either through fractional or integer values that satisfy the equation.

 

[dacruick]

couldnt a be (1,1)/u?

 

[Jack Bateman]

I am not quite sure what you mean. The values of a and b must be scalars.

 

[dacruick]

to be honest I've never seen the <> notation but i assume that means a vector. But how can you add two scalars to get a vector?

 

[Jack Bateman]

This particular notation denotes a vector in component form. The initial point is the origin. The point given indicates the terminal point.

 

[Jack Bateman]

Is there a generalized form that I could use to solve for two different scalars? Every system of equations I try to create is fundamentally flawed in some way.

 

[cupcakethief]

Heres a generalized problem that might help. Just plug and chug your numbers. You should be provided with u and w values.

Heres my example
v= <2,1> (here you'd put <1,1>. i just don't feel like resolving it) v=<1,2> (i+2j) and w=<-1,1> (i-j)

therefore v=2i+j (<2,1> corresponds to the <i,j> coeffs)
so set the two sides equal to each other.
2i + j = a(i + 2j) + b(i - j)
2i + j = (a + b)i + (2a - b)j
simplify and you get
2=a+b and 1=2a-b
Solve and you get a=1 and b=1

If you need anymore help email me. :]

 

[HallsofIvy]

Homework Statement

Find a and b such that v= au + bw given that v = <1, 1>

Homework Equations

v = au + bw, v = <1,1>

The Attempt at a Solution

I have tried to solve for the appropriate values via guess and check, but it hasn't worked out. I cannot find a proportion either through fractional or integer values that satisfy the equation.

What are u and w? This makes no sense without knowing that. For some u and w, there will not exist such a and b. Assuming that u and w form a basis for R², there must be unique a and b but they depend on what u and w are.

If, for example, u= <1, 0> and w= <0, 1> then a= b= 1. If u= <1, 1> and v= <1, -1) then a= 1 and b= 0.

If you are supposed to give some general formula, then let $u= <u_1, u_2>$ and $w= <w_1, w_2>$ then au+ bw= v becomes $a<u_1, u_2>+ b<w_1, w_2>$$= <au_1+ bw_1, au_2+ bw_2>= <1, 1>$ so we must have $au_1+ bw_1= 1$ and $au_2+ bw_2= 1$. Solve those equations for a and b. Of course, the solution will involve a denominator of $u_1w_2- u_2w_1$. There will be a solution if and only if that is not 0, exactly the condition that u and w form a basis.