# Vector equation!!!

• Sep 8th 2008, 08:46 PM
Snowboarder
Vector equation!!!
Hi everyone. I'm having some problems with this couple examples.

Let OAB be a triangle, that is, O,A and B are not collinear. Now let R and S be the mid-points of the sides AB and OA respectively and let M be the point of intersection of the line segments OR and BS.

a)
Express the vector OS as a linear combination of OA and OB.

Is it : OS = OB - 1/2(OA) ??

b)
Express the vector OR as a linear combination of OA and OB.

Is it: OR = OB + 1/2(BA) = OB + 1/2(OA - OB) = 1/2(OA + OB)??

c)

Give the vector equation of the line through O and R in terms of OA and OB.

Is it: OR = OB + BR = OB + tBA???

d)
Give the vector equation of the line through B and S in terms of OA and OB.

e)
Express the vector OM as a scalar multiple of OR.

thanks for any help.

• Sep 9th 2008, 02:24 PM
Laurent
Hi,

Your a) and b) are correct. As for the following, the important is that the equation of a line through $\displaystyle A$ and $\displaystyle B$ may be written geometrically as: $\displaystyle \overrightarrow{AP}=t\overrightarrow{AB}$, $\displaystyle t\in\mathbb{R}$ (meaning that the line is the set of all points $\displaystyle P$ such that ... for some $\displaystyle t\in\mathbb{R}$). You may also write it $\displaystyle P=A+t\overrightarrow{AB}$, $\displaystyle t\in\mathbb{R}$, if you used this notation in your geometry lesson.

So, for c), the line through $\displaystyle O$ and $\displaystyle R$ has the following equation: $\displaystyle \overrightarrow{OP}=t\overrightarrow{OR}=\frac{t}{ 2}(\overrightarrow{OA}+\overrightarrow{OB})$, $\displaystyle t\in\mathbb{R}$.

Try to take it from here. d) works the same, and in e) involves writing that $\displaystyle M$ satisfies both previous equations (with a priori distinct parameters $\displaystyle t$ and $\displaystyle t'$), and eliminating $\displaystyle M$ by combining these equations, in order to find $\displaystyle t$ and $\displaystyle t'$.

Laurent.