9Dgeometry


 * Vectors in Geometry**

Example 1 In the triangle ABC, D is the midpoint of AB and E is the midpoint of AC. Prove that DE is parallel to BC and half the length.

math \lbrace \textit{Aim: show that } \, \overrightarrow{DE}=\frac{1}{2} \overrightarrow{BC} \rbrace \qquad. math

math \\ . \qquad \overrightarrow{DE}=\overrightarrow{DA}+\overrightarrow{AE} \qquad \qquad \textit{by vector addition} \qquad. \\ . \\ . \qquad \overrightarrow{DE}=\frac{1}{2}\overrightarrow{BA}+\frac{1}{2}\overrightarrow{AC} \qquad \textit{since D is midpoint of BA etc} math . math \\ . \qquad \overrightarrow{DE}=\frac{1}{2} \big( \overrightarrow{BA}+\overrightarrow{AC} \big) \qquad. \\ . \\ . \qquad \overrightarrow{DE}=\frac{1}{2}\overrightarrow{BC} \qquad \qquad \qquad \textit{by vector addition} math

Hence DE is parallel to BC and half its length, as required.

Example 2 ABCD is a quadrilateral. E, F, G, H are midpoints of the sides. Prove that EFGH is a parallelogram.

{A parallelogram has opposite sides that are equal and parallel, so:} math . \qquad \lbrace \textit{Aim: show that } \, \overrightarrow{EF}=\overrightarrow{HG} \textit{ and } \overrightarrow{EH}=\overrightarrow{FG} \rbrace \qquad. math

math \\ . \qquad \overrightarrow{EF}=\overrightarrow{EB}+\overrightarrow{BF} \qquad \qquad \textit{by vector addition} \qquad. \\ . \\ . \qquad \overrightarrow{EF}=\frac{1}{2}\overrightarrow{AB}+\frac{1}{2}\overrightarrow{BC} \\. \\ . \qquad \overrightarrow{EF}=\frac{1}{2} \big( \overrightarrow{AB}+\overrightarrow{BC} \big) math . math \\ . \qquad \overrightarrow{EF}=\frac{1}{2}\overrightarrow{AC} \qquad \qquad \qquad \textit{by vector addition} \qquad. \\ . \\ . \qquad \overrightarrow{EF}=\frac{1}{2} \big( \overrightarrow{AD}+\overrightarrow{DC} \big) \qquad \textit{by vector addition} math . math \\ . \qquad \overrightarrow{EF}=\frac{1}{2}\overrightarrow{AD}+\frac{1}{2}\overrightarrow{DC} \qquad. \\ . \\ . \qquad \overrightarrow{EF}=\overrightarrow{HD}+\overrightarrow{DG} \\. \\ . \qquad \overrightarrow{EF}=\overrightarrow{HG} math

Thus EF is parallel and equal to HG.

Using similar logic because the situation is symmetrical, math . \qquad \overrightarrow{EH}=\overrightarrow{FG} math Hence EFGH is a parallelogram, as required.

Example 3 Use the scalar product (dot product) of vectors to prove the **Theorem of Pythagoras**.

From the diagram **__c__** = **__a__** + **__b__**

Also, since **__a__** and **__b__** are perpendicular, math . \qquad \underline{a} \centerdot \underline{b} = 0 \qquad. math math \\ . \qquad c^2=\underline{c} \centerdot \underline{c} \qquad. \\ . \\ . \qquad c^2= \big( \underline{a} + \underline{b} \big) \centerdot \big( \underline{a} + \underline{b} \big) \qquad. \\ . \\ . \qquad c^2=\underline{a} \centerdot \underline{a} + 2\underline{a} \centerdot \underline{b} + \underline{b} \centerdot \underline{b} . \qquad \qquad \qquad \textit{standard binomial expansion} \qquad. math

math \\ . \qquad \text{But } \underline{a} \centerdot \underline{a} =a^2 \text{ and } \underline{a} \centerdot \underline{b} = 0 \qquad. \\ . \\ . \qquad c^2=a^2+2 \times 0 +b^2 \\. \\ . \qquad c^2=a^2+b^2 math

This is the **Theorem of Pythagoras**, as required.

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