Linear Combination of a and b makes c, answer is x flipped over y-axis.

Question

I have been given two lines (a and b) and a line x. I have to make x a linear combination of a and b. My answer, in the preview, is perfect aside from being flipped over the y-axis. I am pretty sure I have done every combination of signs possible however its always wrong. How do I flip (-319/380)a + (49/38)b over the y-axis from Quadrant 1 to Quadrant 3. The vector a is about (0.9 -2) and the vector b is (1 2) and the x is (-2 0.9). Any help would be much appreciated.

Answer 1

You appear to be asking how to find $r$ and $s$ such that

$$r\begin{bmatrix}0.9 \\-2\end{bmatrix}+s\begin{bmatrix}1 \\2\end{bmatrix}=\begin{bmatrix}-2 \\0.9\end{bmatrix}$$

(1) $0.9r+s=-2$

(2) $-2r+2s=0.9$

$2$ x Equation (1) - Equation (2) gives

$3.8r=-4.9$ i.e. $r=-\frac{49}{38}$.

Substituting the value for $r$ into either Equation(1) or (2) gives $s=-\frac{319}{380}$.

Is this what you require?