Linear algebra and vectors. confused

KataKoniK

Let P1 = P1(2, 1, -2) and P2 = P2(1, -2, 0). Find the coordinates of P, which is 1/4 the way from P2 to P1.

The answer is P = P(5/4, -5/4, -1/2)

Can someone explain to me the concept of this? I don't quite understand it. So basically we are suppose to find a vector P that is 1/4 away from (P1 - P2)? The solution manual does (1/4)P1 + (3/4)P2 = P. Why?

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HackaB

The coordinates of points P1 and P2 can also be thought of as the coordinates of vectors extending from the origin to those points. For example, (2, 1, -2) are the coordinates of the vector $$\vec{P_1}$$, and (1, -2, 0) are the coordinates of the vector $$\vec{P_2}$$. First, you need to find the vector A such that

$$\vec{P_2} + \vec{A} = \vec{P_1}$$

Remember that the geometric interpretation of vector addition is adding directed line segments tail to head. To add vector 1 to vector 2, you place the tail of vector 2 on the head of vector 1. The sum is the vector that extends from the tail of 1 to the head of 2. So to find the point 1/4 of the way along the line connecting point P2 to point P1, you need to find the vector

$$\vec{P_2} + \frac{1}{4}\vec{A}$$

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KataKoniK

Thanks for the help

heman

the way which i would have had done it is....

I would have assumed an line segment between those two point and used the ration formula to obtain the coordinates...that seemed to be most simpler

keeti

i am happy with heman 's answer but he should somewhat clear about this answer. the distance between these points is sqrt.now let the p point is 1/4 distance away from P1.them p point cuts that joining line between (2 ,1,-2) and (1,-2,0) in two parts sqrt/4 and 3 *sqrt/4 unit means the ratio will be 1,3
now apply ratio formula. m=1,n=3
X= (m*X1+n*X2)/(m+n)
same for y and z
i think now it will be very clear to katakonik

Galileo

Homework Helper
The equation of a line joining the point given by vectors v and w can be given by: r(t)=(1-t)v+wt

As you can see: r(0)=v and r(1)=w. If you see the geometric picture decribing this line, you can easily see that the point 1/4 the way from the point given by v to the point given by w is r(1/4).

So for your problem: v=(1,-2,0), w=(2,1,-2) and r(1/4)=3/4(1,-2,0)+1/4(2,1,-2).

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