Mathematics

Mathematics

Problem

The points P(3,−1,4)P(3,-1,4) and Q(9,5,−2)Q(9,5,-2) are the ends of a line segment in space. (a) Find the coordinates of the point RR that divides PQPQ internally in the ratio 2:32:3. (b) Calculate the distance OROR, where OO is the origin. (c) Determine the vector PR⃗\vec{PR} in component form.
Intuitive explanation

This problem is about using the section formula to locate a point that divides a line segment in a given ratio, then finding distances and vectors from that point. The key insight is that when a point divides a segment internally in ratio m:nm:n, you're finding a weighted average of the endpoints' coordinates—closer to the endpoint with the larger weight. Once you have RR, the rest follows from straightforward distance and vector calculations.

Intuitive explanation

This problem is about using the section formula to locate a point that divides a line segment in a given ratio, then finding distances and vectors from that point. The key insight is that when a point divides a segment internally in ratio m:nm:n, you're finding a weighted average of the endpoints' coordinates—closer to the endpoint with the larger weight. Once you have RR, the rest follows from straightforward distance and vector calculations.

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