Question 3.2: Show that the line segment joining the midpoints of two side...

We refer to Figure 2. From the figure we see that

But $\overrightarrow{P T}=\frac{1}{2} \overrightarrow{P R}$ and $\overrightarrow{P S}=\frac{1}{2} \overrightarrow{P Q}$. Thus

Hence from Theorem 1.2.3,

Theorem 1 For any vectors $\mathbf{u}, \mathbf{v}, \mathbf{w}$, and scalar $\alpha$,
(i) $\mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}$
(ii) $(\mathbf{u}+\mathbf{v}) \cdot \mathbf{w}=\mathbf{u} \cdot \mathbf{w}+\mathbf{v} \cdot \mathbf{w}$
(iii) $(\alpha \mathbf{u}) \cdot \mathbf{v}=\alpha(\mathbf{u} \cdot \mathbf{v})$
(iv) $\mathbf{u} \cdot \mathbf{u} \geq 0$; and $\mathbf{u} \cdot \mathbf{u}=0$ if and only if $\mathbf{u}=\mathbf{0}$

Theorem 2 Let u and v be two nonzero vectors. Then if φ is the angle between them, $\cos \varphi=\frac{\mathbf{u}^{\bullet} \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}$

Theorem 3 If u ≠ 0, then v = αu for some nonzero constant a if and only if u and v are parallel.

$\overrightarrow{S T}$ is parallel to $\overrightarrow{Q R}$ Moreover $|\overrightarrow{S T}|=\frac{1}{2}|\overrightarrow{Q R}|$. This is what we wanted to show.