Vector Geometry Find the ratio in which a points divides two lines segments in a triangle

In the given triangle OAB, C is the mid-point and D divides OA in the ratio of 3:1. Point P is on the intersection of OC and BD, Find the ratio in which P divides OC and BD.
Q12. Give coordinates of points A(2, -1, 2) and B(2, -2, 5).
Find position vector of the point P which divides the segment AB in the ratio of 1:2.

[PrithwirajSen-njqq]

First we may express
1) vector AB
vector OC
& vector BD
wrt vector b and vector OA.
Then we may take vector OP = y fraction of OC
& vector BP =x fraction of vector BD
Then vector OP and vector BP both may be expressed wrt
vector b and vector OA
As a result vector OP may be expressed in two ways and both are equal.
Then on equating the coefficients of vector
b in the two forms
and equating the coefficients of vector OA in the two forms
we may get two equations of x and y wherefrom we will get the value of x and y.
OP =y then PC =1-y
Then we will get the ratio of OP and PC
BP =x then PD =1- x
Then we will get the ratio of BP and PD

[studyaccount]

For anyone wondering, is there any method then using 1-t and t, 1-s and s ....instead the better logic would be to set the intersecting point of the OP to be either lambda or mu (set 1 equation consisting of lambda, and the other with mu)....hope this makes sense...if u dont understand in the future, just comment here and i will explain

[jasmanc]

why the length of OC is 1?, I don´t understand! :(

[cheroivy]

Nice explanation I understood very well

[simontemplar]

Thankyou. Where is the link to the answer please?

[Warlord_Megatron]

Was missing on the part on how to get the values as i kept forgetting that we have to compare the coefficients. Thanks!

[KazACWizard]

is it just me or am i too harsh? i expect to already know this, but i dont seem to think that i can understand this... help plz :(

[rosalinlazaroff]

Great Video! Please can you provide a solution to the problem at the end? Thanks

[giorgioivanho]

can I ask why OP is t and PC is 1-t I can't seem to understand that

[ashantharaka]

How can write OP=t *OC ???
Is it not OP=t/(1-t) * OC

[reallifemathsbyanshuman]

Sir aap iitjam Ka bhi karaye agar time ho please

[rosalinlazaroff]

I got 25:11...just does NOT seem correct!