I'm new to vectors and need to answer the following two things:
- Show by using vectors that the mid segment of a trapezoid is parallel to the bases and $1/2$ as long as the sum of them.
That's what I got so far.
- Show by using vectors that the line segment m in a quadrilateral is equal to the sum of both not divided side-vectors divided by two.
The work sheet only provides the solutions but I can't figure out how to get there.
Thanks!
We do not work with free vectors. We work with position vectors, which are fixed on the Euclidean plane wrt some origin. (See figure.) The underlying coordinate plane provides coordinates to any points $P,Q$. This facilitates writing $\vec{OP}=P-O$ and $\vec{PQ}=Q-P$. For example if $P=(3,4)$ on coordinate plane, then $\vec{OP}=(3-0,4-0)=3\hat{i}+4\hat{j}$.
So let position vectors of the vertices of trapezium be $A,B,C,D$. Since $E$ is midpoint of $AD$, its position vector will be (think coordinates) $$E=\frac{A+D}{2} \, , \quad F=\frac{B+C}{2}$$
Now lengths are given by magnitudes of vectors $$|AB|=|\vec{AB}|=B-A \, , \quad |DC|=|\vec{DC}|=C-D$$
Since $AB // DC$, $\,\vec{DC}=k\vec{AB}$.
Now $$|EF|=F-E=\frac{(B-A)+(C-D)}{2}=\frac{1}{2}(|AB|+|DC|)$$
You can also conclude about parallelism.
This was Ques $1$. Now you can do Ques $2$.