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ABCD is a parallelogram. The diagonals of ABCD intersect at O. OA =a and OB=b.
a) Express the vector CA in terms of a
b) Express the vector AB in terms of a and b
c) Express the vector BC in terms of a and b
(Mathswatch, Vectors Introduction)

Sagot :

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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:

  1. [tex]OA \cong OC[/tex]
  2. [tex]OB \cong OD[/tex]

Hence, we have the following vectorial expressions:

  1. [tex]\overrightarrow{OA} = \vec a[/tex] (1)
  2. [tex]\overrightarrow{OB} = \vec b[/tex] (2)
  3. [tex]\overrightarrow{OC} = -\vec {a}[/tex] (3)
  4. [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].

To learn more on vectors, we kindly invite to check this verified question: https://brainly.com/question/21925479

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