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

[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) $$\overrightarrow{CD} $$

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) $$\overrightarrow{AD}$$

 

[soroban]

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) $$\overrightarrow{AD}$$

$$\overrightarrow{AD} \;=\;\overrightarrow{AO} + \overrightarrow{OD} \;=\;-\overrightarrow{OA} + \overrightarrow{OD} \;=\;\overrightarrow{OD} - \overrightarrow{OA}$$

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

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