Looking for the Point in 4-dimensional space

Geometry Level 3

Let $P(1,2,11, 0)$ and $Q(5, -2, -1, 16)$ be two points in a 4-dimensional coordinate system. Furthermore, the distance between point $X(x,y,z,w)$ and $P$ is three times the distance between $Q$ and $X$ .

If $P, Q$ and $X$ are collinear, then find $x + y + z + w$ .

The answer is 17.

1 solution

Christoph Pi
Mar 28, 2017

The point X we're looking for is a weighted arithmetic average: X=3*Q+P. Therefore, we simply have to calculate the point X = (4, -1, 2, 12), and the sum of the coordinate values is 4-1+2+12=17

It should be mentioned that X lies on the "line" joining P and Q, otherwise the coordinates of X will not be unique.

Edit: As a moderator, I've added that P,Q,X must be collinear so that others don't start reporting the question. I hope that is ok with you. I'm not actually certain if the term "collinear" is appropriate in 4-dimension, but I think people will understand what is intended by its use.

Brian Charlesworth - 4 years, 2 months ago

I understood what it meant. I suppose you could also say that the vector from P to Q is a scalar multiple of the vector from P to X.

Steven Chase - 4 years, 2 months ago

Ok, great. That's a good suggestion referring to vectors rather than lines, but I think I'll let Christoph decide on any further changes to his question.

Brian Charlesworth - 4 years, 2 months ago

Looking for the Point in 4-dimensional space

The answer is 17.

1 solution

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