smc-1211-20-Rhombus

Vectors, EXT1 V1 EQ-Bank 27

`OABC` are the vertices of a rhombus.

Prove, using vectors, that its diagonals are perpendicular. (2 marks)

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`text(See Worked Solution)`

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`text(S) text(ince)\ \ OABC\ \ text(is a rhombus)`

`| underset~a | = | underset~c |`

`text(Diagonals are) \ \ overset(->)(OB) \ \ text(and) \ \ overset(->)(AC) , text(where)`

`overset(->)(OB) = underset~a + underset~c`

`overset(->)(AC) = underset~c – underset~a`

`(underset~c – underset~a) · (underset~a + underset~c)`	`= underset~c* underset~a + underset~c* underset~c – underset~a* underset~a – underset~a *underset~c`
	`=\| underset~c \|^2 – \| underset~a \|^2`
	`= 0`

`:. \ overset(->)(OB) ⊥ overset(->)(AC)`

Vectors, EXT1 V1 SM-Bank 22

Consider the rhombus `OPQR` shown below, where `overset(->)(OP) = punderset~i` and `overset(->)(OR) = underset~i + 2underset~j` and `p` is a real constant.

Find `p`. (1 mark)

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Show that the diagonals of rhombus `OPQR` are perpendicular. (2 marks)

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`sqrt5`
`text(See Worked Solutions)`

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i. `|overset(->)(OP)| = |overset(->)(OR)|`

`p`	`= sqrt(1^2 + 2^2)`
	`= sqrt5`

ii. `overset(->)(RQ) = overset(->)(OP),\ \ overset(->)(OR) = overset(->)(PQ)`

`overset(->)(OQ)`	`= overset(->)(OR) + overset(->)(RQ)`
	`= [(1),(2)] + [(sqrt5),(0)]`
	`= (1 + sqrt5)underset~i + 2underset~j`

`overset(->)(PR)`	`= overset(->)(OR) – overset(->)(RQ)`
	`= [(1),(2)] – [(sqrt5),(0)]`
	`= (1 – sqrt5)underset~i + 2underset~j`

`overset(->)(OQ) · overset(->)(PR)`	`= [(1 + sqrt5),(2)] · [(1 – sqrt5),(2)]`
	`= (1 + sqrt5)(1 – sqrt5) + 2 xx 2`
	`= 1 – 5 + 4`
	`= 0`

`:. overset(->)(OQ) ⊥ overset(->)(PQ)`

Vectors, EXT1 V1 2011 VCE 10 MC

The diagram below shows a rhombus, spanned by the two vectors `underset~a` and `underset~b`.

It follows that

`underset~a*underset~b = 0`
`underset~a = underset~b`
`(underset~a + underset~b)*(underset~a-underset~b) = 0`
`|\ underset~a + underset~b\ | = |\ underset~a-underset~b|`

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`C`

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`text(Consider A:)`

`text(If)\ \ underset~a · underset~b = 0\ \ =>\ \ underset~a ⊥ underset~b\ \ text{(incorrect)}`

`text(Consider B:)`

`underset~a != underset~b\ \ text{(incorrect)}`

`text(Consider C:)`

`underset~a + underset~b`	`= overset(->)(OC)`
`underset~a-underset~b`	`= overset(->)(BA)`

`overset(->)(OC) · overset(->)(BA)=0`

`text{The diagonals of a rhombus are perpendicular (correct)}`

`=> C`