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

 

[Raanikeri]

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

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 [imath]\mathbf a[/imath] and [imath]\mathbf b[/imath] are linearly independent, at least in the drawing, you know that if [imath]p\mathbf a + q\mathbf b = r\mathbf a+s\mathbf b[/imath] then [imath]p = r[/imath] and [imath]q=s[/imath].

 

[Jim77]

Did you get the solution?

 

[Raanikeri]

Did you get the solution?

yeah it's 2b