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

[karush]

View attachment 1019\(\displaystyle ABCD\) is a rectangle and \(\displaystyle O\) is the midpoint of \(\displaystyle [AB]\).

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

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 \(\displaystyle \overrightarrow{CD} = 0\)

(b) \(\displaystyle \overrightarrow{OA}\)
(c) \(\displaystyle \overrightarrow{AD}\)

 

[soroban]

Re: vectors inside a rectangle

Hello, karush!

View attachment 1019

\(\displaystyle ABCD\) is a rectangle and \(\displaystyle O\) is the midpoint of \(\displaystyle [AB]\).

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

(a) \(\displaystyle \overrightarrow{CD} \)

$$\overrightarrow{CD} \;=\;\overrightarrow{CO} + \overrightarrow{OD} \;=\;-\overrightarrow{OC} + \overrightarrow{OD} \;=\;\overrightarrow{OD} - \overrightarrow{OC}$$

(b) \(\displaystyle \overrightarrow{OA}\)

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

(c) \(\displaystyle \overrightarrow{AD}\)

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

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

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