Linear Algebra: Two vectors and Unknown Ks.

Question

Here is a problem I cannot get my head around;

The question asks to find the values of K1 and K2 in the following expression:

k1v+k2w, knowing that v=-2i+2.5j and w=[4 -0.5].

The only other information the problem provides is that the sum of k1v+k2w is equal to [-23 -18].

(Edit) As a wonderful person in the comments recommended, here is what I have tried:

I tried to find the resulting vector in terms of i and j, and then isolate the Ks for each, but it doesn’t work when I input them back into the formula.

It gives me K1=-11.5 and k2=-9

I thank you very much for a fast and detailed answer.

Answer 1

Write down the equation $k_1\vec v+k_2\vec w=-23\vec i-18\vec j$ in the two componennts separately and solve the ensuing $2\times 2$ inhomogenous linear system of equations; alternatively multiply that equation with the vector dot product once by $\vec i$ and once by $\vec j$, assuming $\vec i\perp \vec j$ such that their dot product vanishes to get $k_1$ and $k_2$ directly.

Answer 2

-2It's odd that the vectors are written in two different notations. One vector is given as v= -2i+2.5j which is the same as [-2 2.5], another is given as w=[4 -0.5] which is the same as 4i- 0.5j, and the third is given as [-23 -18] which is the same as -23i- 18j.

So in one notation, $k_1v+ k_2w= -2k_1i+ 2.5k_1j+ 4k_2i- 0.5k_2j= (-2k_1+ 4k_2)i+ (2.5k_1- 0.5k_2)j = -23i- 18j$.

In the other notation,$k_1v+ k_2w= [-2k_1 2.5k_1]+ [4k_2 0.5k_2]= [-2k_1+ 4k_2 2.5k_1+ 0.5k_2]= [-23 -18]$.

In either notation we must have $-2k_1+ 4k_2= -23$ and $2.5k_1+ 0.5k_2= -18. Two equations to solve for two unknowns.