I'm given the following $\text{true}$ or $\text{false}$ statement, where i must provide justification for my answer:
$$\text{The points in the plane corresponding to} \begin{bmatrix}
-2 \\
5 \\
\end{bmatrix} \text{and} \begin{bmatrix}
-5 \\
2 \\
\end{bmatrix} \text{lie on a line through the origin.}$$
My initial inclination is to say $\text{true}$ , and have a little sketch of the two vectors plotted on a graph where the tails of both the given vectors start at the origin, for justification. But feel like i either misunderstand what the question is asking, or that in the $\text{true}$ or $\text{false}$ statement, line is only singular and not plural if that makes a difference? Since the two vectors do form separate lines from the origin and not just one?
Best Answer
You can use the equation for a line of $y=mx+b$
Your slope is $$\frac{y_2-y_1}{x2-x1} = \frac{2-5}{-5-(-2)} = 1$$
So $y = x + b$. Use one of your points. $$5 = -2 + b$$ $$b = 7$$
At this point you know the y-intercept is not 0, so it can't run through the origin. For completeness, plug in the point $(0,0)$ for the origin. $$0 = 0 + 7$$
Doesn't work out, so the statement is false.
As for a diagram, plot the two points. Draw a line between them, and extend that line straight past them as well. If that line doesn't touch $(0,0)$, the statement is false.
In terms of vectors, they would need to be parallel or antiparallel.