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.