- #1

EmilyHopkins

- 8

- 0

## Homework Statement

The point P and Q have postion vectors a + b, and 3a - 2b respectively, relative to the origin O.Given that OPQR is a parallelogram express the vector PQ and PR in terms of a and b. By evaluating two scalar products show that if OPQR is a square then |a |

^{2}= 2 |b |

^{2}

## The Attempt at a Solution

OP = a + b

OQ = 3a -2b

PQ = OR= PO + OQ = -OP +OQ = -(a + b) + (3a - 2b) = 2a - 3b

PR = PO + OR = -OP + OR = -(a + b) + (2a - 3b) = a - 4b

So now the question says use two scalar products to show |a |

^{2}= 2 |b |

^{2}. I'm assuming since the question ask for these two vectors in terms of a, and b that you will have to utilize it to get the result. So I drew out the square for a visualization.

So since its a square it means the dot product of

OP.PQ= 0

(a + b)(2a - 3b) = 2a

^{2}-ab -3b

^{2}= 0

ab = 2a

^{2}-3b

^{2}

PR.OQ= 0 (Since their perpendicular)

(a-4b)(3a - 2b) =0

3a

^{2}- 14ab +8b

^{2}=0

3a

^{2}-14(2a

^{2}-3b

^{2}) +8b

^{2}=0

3a

^{2}-28b

^{2}+42a

^{2}+ 8b

^{2}=0

-25a

^{2}+50b

^{2}=0

25a

^{2}=50b

^{2}

a

^{2}= 2b

^{2}

Is their a faster way to work it or is this correct ?

?