# Vector equation!!!

• September 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.

• September 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 $A$ and $B$ may be written geometrically as: $\overrightarrow{AP}=t\overrightarrow{AB}$, $t\in\mathbb{R}$ (meaning that the line is the set of all points $P$ such that ... for some $t\in\mathbb{R}$). You may also write it $P=A+t\overrightarrow{AB}$, $t\in\mathbb{R}$, if you used this notation in your geometry lesson.

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

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

Laurent.