Formula for determining if one vector is a multiple for two others

[robertjford80]

Homework Statement

The Attempt at a Solution

Is there a formula for determining

x = 3v₁ - 2v₂

It looks like you just have to do trial and error. I tried Gaussian Elimination on the two vectors and got -1.5 and .5 so that didn't work.

 

[DryRun]

Do you mean, how to find a1 and a2? Just substitute the given vectors and solve.

 

[Steely Dan]

Quite simply, by comparing the three components of the vector on each side of the equation, the following things must be true:
$$x^1 = a_1 v^1_1 + a_2 v^1_2$$
$$x^2 = a_1 v^2_1 + a_2 v^2_2$$
$$x^3 = a_1 v^3_1 + a_2 v^3_2$$
where the superscript denotes the first, second or third element of the vector. This leads to the set of equations
$$-1 = a_1 + 2 a_2$$
$$-3 = -a_1$$
$$4 = 2a_1 + a_2$$
From there it should be obvious how that solution was obtained.

 

[robertjford80]

I mean how they found out that 3 times vector 1 - 2 times vector 2 = x = (-1,-3,4)

 

[robertjford80]

Steely Dan, thanks, I got it.

when i did gaussian elimination i wrote the vector horizontally rather than vertically, that's why I was wrong.