Vector question

[Raanikeri]

Hi, I'm not really understanding the solution for this question.
I can get:

but then I don't know how to proceed. I know that BE is a multiple of b because BE is parallel to OE... but how do I find what the multiple is from the information I have?

 

[blamocur]

You know that $\overrightarrow{OD} = x\cdot\mathbf b$, where $x$ is unknown at this point. Can you now express the fact that $C,D,E$ lie on the same line? Hint: $\overrightarrow{DE} = t\cdot\overrightarrow{CD}$ for some $t$.

 

[Raanikeri]

You know that $\overrightarrow{OD} = x\cdot\mathbf b$, where $x$ is unknown at this point. Can you now express the fact that $C,D,E$ lie on the same line? Hint: $\overrightarrow{DE} = t\cdot\overrightarrow{CD}$ for some $t$.

yes but then there are 2 different scale factor amounts now, right? the scale factor x for lines parallel to OD and the scale factor t for lines parallel to DE.. How can I find 2 unknowns - x and t - when I can't form simultaneous equations to solve them?

 

[blamocur]

yes but then there are 2 different scale factor amounts now, right? the scale factor x for lines parallel to OD and the scale factor t for lines parallel to DE.. How can I find 2 unknowns - x and t - when I can't form simultaneous equations to solve them?

Vector equations are equivalent to 2 scalar equations -- one per coordinate. Since $\mathbf a$ and $\mathbf b$ are linearly independent, at least in the drawing, you know that if $p\mathbf a + q\mathbf b = r\mathbf a+s\mathbf b$ then $p = r$ and $q=s$.

 

[Jim77]

Did you get the solution?

 

[Raanikeri]

Did you get the solution?

yeah it's 2b