Question

13 in the diagram, abcdef is a regular hexagon, ( overrightarrow{oa}=mathbf{a} ), and ( overrightarrow{ob}=mathbf{b} ). find in terms of ( mathbf{a} ) and ( mathbf{b} ): (a) ( overrightarrow{ad} ) (b) ( overrightarrow{fa} ) (c) ( overrightarrow{da} ) (d) ( overrightarrow{ab} ) (e) ( overrightarrow{cf} ) (f) ( overrightarrow{de} )

Explanation:

(a) Step1: Analyze the position of vectors in a regular hexagon

In a regular hexagon \(ABCDEF\), \(OA = a\), and \(A\) and \(D\) are opposite vertices. Since \(O\) is the center of the hexagon, \(\overrightarrow{AD}\) is in the opposite direction of \(\overrightarrow{OA}\) and has twice the length.

(a) Step2: Calculate \(\overrightarrow{AD}\)

\(\overrightarrow{AD}=\overrightarrow{AO}+\overrightarrow{OD}\), and \(\overrightarrow{AO}=-\mathbf{a}\), \(\overrightarrow{OD}=-\mathbf{a}\), so \(\overrightarrow{AD}=- 2\mathbf{a}\)

(b) Step1: Use vector subtraction

\(\overrightarrow{FA}=\overrightarrow{OA}-\overrightarrow{OF}\), and in a regular hexagon \(\overrightarrow{OF}=\mathbf{b}\)

(b) Step2: Calculate \(\overrightarrow{FA}\)

\(\overrightarrow{FA}=\mathbf{a}-\mathbf{b}\)

(c) Step1: Analyze the relationship with \(\overrightarrow{AD}\)

\(\overrightarrow{DA}\) is the opposite vector of \(\overrightarrow{AD}\)

(c) Step2: Calculate \(\overrightarrow{DA}\)

Since \(\overrightarrow{AD}=-2\mathbf{a}\), then \(\overrightarrow{DA}=2\mathbf{a}\)

(d) Step1: Use vector subtraction

\(\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}\)

(d) Step2: Calculate \(\overrightarrow{AB}\)

Given \(\overrightarrow{OA}=\mathbf{a}\), \(\overrightarrow{OB}=\mathbf{b}\), so \(\overrightarrow{AB}=\mathbf{b}-\mathbf{a}\)

(e) Step1: Analyze the position of vectors

\(C\) and \(F\) are related to the center \(O\). \(\overrightarrow{CF}=\overrightarrow{CO}+\overrightarrow{OF}\), and \(\overrightarrow{CO}=-\mathbf{b}\), \(\overrightarrow{OF}=-\mathbf{b}\)

(e) Step2: Calculate \(\overrightarrow{CF}\)

\(\overrightarrow{CF}=-2\mathbf{b}\)

(f) Step1: Use the property of a regular hexagon

\(\overrightarrow{DE}\) and \(\overrightarrow{FA}\) are equal vectors (parallel and same - length)

(f) Step2: Calculate \(\overrightarrow{DE}\)

Since \(\overrightarrow{FA}=\mathbf{a}-\mathbf{b}\), then \(\overrightarrow{DE}=\mathbf{a}-\mathbf{b}\)

Answer:

(a)

$$\overrightarrow{AD}=-2\mathbf{a}$$

(b)

$$\overrightarrow{FA}=\mathbf{a}-\mathbf{b}$$

(c)

$$\overrightarrow{DA}=2\mathbf{a}$$

(d)

$$\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$$

(e)

$$\overrightarrow{CF}=-2\mathbf{b}$$

(f)

$$\overrightarrow{DE}=\mathbf{a}-\mathbf{b}$$