Question

A sum of scalar multiples of two vectors (such as au + bv, where a and b are scalars) is called a linear conbination of the vectors.Let u= <3,3> and v= <-3,3>. Express <3,-27> as a linear combination of u and v.

Answer 1

-4u - 5v = <3, -27>

Explanation:

u = <3,3> and v = <-3,3>

expressing the problem as a linear combination with scalars a & b

au + bv = <3,27>

a<3,3> + b<-3,3> = <3,-27> (multiplying the scalar terms into the vectors)

<3a,3a> + <-3b,3b> = <3,-27>

we can separate the vectors into their vertical and horizontal components.

Equating the horizontal components of the vector:

3a - 3b = 3

or

a - b = 1 -----> eq 1

Equating the vertical components of the vector:

3a + 3b = -27

or

a + b = -9 -----> eq 2

Now we have 2 variables and 2 equations, solving system of equations:

by elimination: eq 1 + eq 2, we get

2a = -8

a = -4

substitute this back into equation 1,

we get b = -5

hence assembling the equation

a<3,3> + b<-3,3> = <3,-27>

-4 <3,3> -5 <-3,3> = <3,-27>

or

-4u - 5v = <3, -27> (answer)

Answer 2

-4u -5v

Explanation:

Let the sum be ...

<3, -27> = a<3, 3> +b<-3, 3>

This resolves to two equations

3 = 3a -3b

-27 = 3a +3b

Adding these together, we get

-24 = 6a

a = -4

Substituting into the second equation gives ...

-27 = 3(-4) +3b

-15 = 3b

-5 = b

The desired linear combination is ...

<3, -27> = -4<3, 3> -5<-3, 3> = -4u -5v