Answer :
a) The vector [tex]\overrightarrow{CA}[/tex] is represented by [tex]2\cdot \vec a[/tex].
b) The vector [tex]\overrightarrow{AB}[/tex] is represented by [tex]\vec b - \vec a[/tex].
c) The vector [tex]\overrightarrow{BC}[/tex] is represented by [tex]\vec a + \vec b[/tex].
Procedure - Vectors in a parallelogram
a) Vector [tex]\overrightarrow {CA}[/tex] in terms of [tex]\vec a[/tex]
By geometry we know that diagonals in a parallelogram fulfill the following properties:
- [tex]OA \cong OC[/tex]
- [tex]OB \cong OD[/tex]
Hence, we have the following vectorial expressions:
- [tex]\overrightarrow{OA} = \vec a[/tex] (1)
- [tex]\overrightarrow{OB} = \vec b[/tex] (2)
- [tex]\overrightarrow{OC} = -\vec {a}[/tex] (3)
- [tex]\overrightarrow {OD} = -\vec b[/tex] (4)
Hence, we have the following vectorial expression for CA:
[tex]\overrightarrow{CA} = 2\cdot \vec a[/tex] (5)
The vector [tex]\overrightarrow{CA}[/tex] is represented by [tex]2\cdot \vec a[/tex]. [tex]\blacksquare[/tex]
b) Vector [tex]\overrightarrow{AB}[/tex] in terms of [tex]\vec a[/tex] and [tex]\vec b[/tex]
The vector [tex]\overrightarrow {AB}[/tex] is defined by the following expression:
[tex]\overrightarrow{OA} + \overrightarrow{AB} = \overrightarrow{OB}[/tex]
[tex]\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow {OA}[/tex] (6)
By (1) and (2):
[tex]\overrightarrow{AB} = \vec b - \vec a[/tex] (7)
The vector [tex]\overrightarrow{AB}[/tex] is represented by [tex]\vec b - \vec a[/tex]. [tex]\blacksquare[/tex]
c) Vector [tex]\overrightarrow{BC}[/tex] in terms of [tex]\vec a[/tex] and [tex]\vec b[/tex]
The vector [tex]\overrightarrow{BC}[/tex] is defined by the following expression:
[tex]\overrightarrow{BC} = \overrightarrow{CA}+\overrightarrow{AB}[/tex] (8)
By (5) and (7):
[tex]\overrightarrow{BC} = 2\cdot \vec a + (\vec b - \vec a)[/tex]
[tex]\overrightarrow{BC} = \vec a + \vec b[/tex] (9)
The vector [tex]\overrightarrow{BC}[/tex] is represented by [tex]\vec a + \vec b[/tex]. [tex]\blacksquare[/tex]
Remark
The statement presents mistakes. Correct form is presented below:
ABCD is a parallelogram. The diagonals of ABCD intersect at O. [tex]\overrightarrow{OA} = \vec a[/tex] and [tex]\overrightarrow{OB} = \vec b[/tex].
a) Express the vector [tex]\overrightarrow {CA}[/tex] in terms of [tex]\vec a[/tex].
b) Express the vector [tex]\overrightarrow {AB}[/tex] in terms of [tex]\vec a[/tex] and [tex]\vec b[/tex].
c) Express the vector [tex]\overrightarrow {BC}[/tex] in terms of [tex]\vec a[/tex] and [tex]\vec b[/tex].
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