MHB How Do You Express Vectors in a Rectangle Using Midpoint Coordinates?

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In the discussion, participants explore expressing vectors in a rectangle, specifically using midpoint coordinates. For vector \(\overrightarrow{CD}\), it is derived as \(\overrightarrow{OD} - \overrightarrow{OC}\). The vector \(\overrightarrow{OA}\) is expressed as half of \(\overrightarrow{CD}\), resulting in \(\frac{1}{2}(\overrightarrow{OD} - \overrightarrow{OC})\). Lastly, \(\overrightarrow{AD}\) is calculated to be \(\frac{1}{2}(\overrightarrow{OD} + \overrightarrow{OC})\). The discussion emphasizes the relationships between these vectors in the context of the rectangle's geometry.
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View attachment 1019$$ABCD$$ is a rectangle and $$O$$ is the midpoint of $$[AB]$$.

Express each of the following vectors in terms of $$\overrightarrow{OC}$$ and $$\overrightarrow{OD}$$
(a) $$\overrightarrow{CD} $$

ok I am fairly new to vectors and know this is a simple problem but still need some input
on (a) I thot this would be a vector difference but this would make $$\overrightarrow{CD} = 0$$

(b) $$\overrightarrow{OA}$$
(c) $$\overrightarrow{AD}$$
 
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Re: vectors inside a rectangle

Hello, karush!

View attachment 1019

$$ABCD$$ is a rectangle and $$O$$ is the midpoint of $$[AB]$$.

Express each of the following vectors in terms of $$\overrightarrow{OC}$$ and $$\overrightarrow{OD}$$

(a) $$\overrightarrow{CD} $$
\overrightarrow{CD} \;=\;\overrightarrow{CO} + \overrightarrow{OD} \;=\;-\overrightarrow{OC} + \overrightarrow{OD} \;=\;\overrightarrow{OD} - \overrightarrow{OC}
(b) $$\overrightarrow{OA}$$
\overrightarrow{OA} \;=\;\tfrac{1}{2}\overrightarrow{CD} \;=\;\tfrac{1}{2}\left(\overrightarrow{OD} - \overrightarrow{OC}\right)
(c) $$\overrightarrow{AD}$$
\overrightarrow{AD} \;=\;\overrightarrow{AO} + \overrightarrow{OD} \;=\;-\overrightarrow{OA} + \overrightarrow{OD} \;=\;\overrightarrow{OD} - \overrightarrow{OA}

. . . .=\;\overrightarrow{OD} - \tfrac{1}{2}\left(\overrightarrow{OD} - \overrightarrow{OC}\right) \;=\;\overrightarrow{OD} - \tfrac{1}{2}\overrightarrow{OD} + \tfrac{1}{2}\overrightarrow{OC}

. . . .=\;\tfrac{1}{2}\overrightarrow{OD} + \tfrac{1}{2}\overrightarrow{OC} \;=\;\tfrac{1}{2}\left(\overrightarrow{OD} + \overrightarrow{OC}\right)
 
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