# Determining if position vectors are on a line through the origin

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?

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I think it's just asking if the line through the points (x,y)=(-2,5) and (-5,2) goes through the origin. Answer seems to be no. One way to show this would be to solve for the line and show that x=0,y=0 does not satisfy the equation. Another way to do it would be to show that the two position vectors are not collinear... they are not parallel or antiparallel. For two points to lie on a line through the origin, the two position vectors need to be collinear... so the two vectors connected together form a line through the origin. – Ameet Sharma Apr 25 '14 at 3:38

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.

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The two position vectors would have to be parallel or antiparallel... one is a constant times the other. – Ameet Sharma Apr 25 '14 at 3:41
@AmeetSharma Thanks for the catch. What I pictured in my head and the word that came out were out of sync. – RandomUser Apr 25 '14 at 3:43
So basically because line was not plural in the given statement, its false? Although it kind of sounds odd, if they had said: "The points in the plane corresponding to $\vec v_1$ and $\vec v_2$ lie on $\mathbf{lines}$ through the origin. Then this statement would be true? Since i mean they do form two separate lines that touch $(0,0)$? – Finding Nemo 2 is happening. Apr 25 '14 at 3:52
Since the vectors have to lie on the line rather than cross the line, then each must run along the same line. The only way for each vector to run completely along the same line is for them to point in the same or opposite directions. When thinking of points, asking if they lie on a line that goes through the origin is the way to word it. With vectors you could say if they lie on the same line. They're functionally equivalent. – RandomUser Apr 25 '14 at 4:03