Maths Vector Question

symx SG

Do kindly help me in this question, thank you.

skythewood SG

OC = OB + BC
= OB + 3OA (BC = 3OA)

AP = 1/4 AB (given)
= 1/4 (AO + OB)
= 1/4 (OB -OA)

OP = OA + AP

ii)

O, P and C together forms a straight line

iii)

find |PA|, and |BA|.

area of triangle is 1/2 x base x height.

for OAP, 1/2 x |PA| x height

for OBP, 1/2 x |BA| x height

for both triangle, the height is the same

XxHoWxX SG

i)(a) OC= OB +BC

= b + 3 OA

= b + 3a

i)(b) AP = 1/4 (OB - OA)

= 1/4 (b - a)

i)(c) OP = OA + AP

= a + 1/4 (b - a)

= a + 1/4 b - 1/4 a

= 1/4 (3a + b)

ii) OPC is a straight line as OP is 1/4 of OC.

iii) I do not know how to do this.

symx SG

AP = 1/4 AB (given)
= 1/4 (AO + OB)
= 1/4 (OB -OA)

Why AO must invert into OA? Is it because value have to be read off from the origin which is O.

skythewood SG

directions is important for vectors. AO = -OA.

note the negative sign...

OA = a

AO = -a

symx SG

Oh I c, if values must read off from origin and by doing so for AO. AO will need to be inverted and thus the value become negative. Am I right?

skythewood SG

no.

the arrow thingie on the diagram says OA is vector a.

OA means go from O to A.

if you go from A to O, the vector is -a

not because of origins.

symx SG

Ok, I understand some parts but I still have quesiton on this segments.

symx SG

OP = OA + AP

I don't get the answer which is 1/4(b+3a) and also for part 3 the answer is 1/3. What value should I sub into |PA| and |BA|?

skythewood SG

AP = 1/4 (OB -OA), from 1B)

OP = OA + AP
   = OA + 1/4(OB - OA)
   = OA + 1/4OB - 1/4OA
   = 3/4OA + 1/4OB
   = 1/4( 3OA + OB)
   = 1/4 (3a + b) (from diagram, OA = a, OB = b)

skythewood SG

|PA| means modulus of the vector PA. this mean find it's length.of vector PA. direction is no longer important as either direction will have the same length.

length of AP, |AP| = |1/4(b-a)|
= 1/4 |b-a|

BP = BO + OP
   = -OB + OP
   = -b + 1/4(3a + b)
   = -b +3/4a +1/4b
   = 3/4a - 3/4b

|BP| = |3/4a - 3/4b|
=3/4 |a-b|
= 3/4 |b-a| |b-a| = |a-b|, since direction don't matters

hence |BP| is 3 times bigger than |AP|

symx SG

Thank you very much for your help.