Point $P$ partitions the directed segment from $A$ to $B$ into a 1:3 ratio. $Q$ partitions the directed segment from $B$ to $A$ into a $1: 3$ ratio. Are P and Q the same point? Why or why not?
A. Yes, they both partition the segment into a 1:3 ratio.
B. Yes, they are both $\frac{1}{4}$ the distance from one endpoint to the other.
C. No, $P$ is $\frac{1}{4}$ the distance from $A$ to $B$, and $Q$ is $\frac{1}{4}$ the distance from $B$ to $A$.
D. No, Q is closer to A and P is closer to B.
Answer (1)
Point P divides the segment A B in a 1 : 3 ratio, so its position vector is p = 4 3 a + b .
Point Q divides the segment B A in a 1 : 3 ratio, so its position vector is q = 4 a + 3 b .
Comparing the position vectors, p = q unless A and B are the same point.
Therefore, P and Q are not the same point: N o , P is 4 1 the distance from A to B , and Q is 4 1 the distance from B to A .
Explanation
Problem Analysis Let A and B be two distinct points in space. We can represent these points using position vectors a and b , respectively. The problem states that point P divides the directed segment from A to B in a 1 : 3 ratio, and point Q divides the directed segment from B to A in a 1 : 3 ratio. We need to determine if P and Q are the same point.
Calculating Position Vectors To find the position vector p of point P , we use the section formula for directed segments. Since P divides A B in the ratio 1 : 3 , we have: p = 1 + 3 3 a + 1 b = 4 3 a + b Similarly, to find the position vector q of point Q , we use the section formula for directed segments. Since Q divides B A in the ratio 1 : 3 , we have: q = 1 + 3 3 b + 1 a = 4 a + 3 b
Comparing Position Vectors Now, we compare the position vectors p and q :
p = 4 3 a + b q = 4 a + 3 b In general, p = q , because 3 a + b is not necessarily equal to a + 3 b . They are only equal if a = b , which means A and B are the same point. However, we are given that A and B are distinct points.
Conclusion Since p = q , points P and Q are not the same. Point P is located 4 1 of the distance from A to B , while point Q is located 4 1 of the distance from B to A . Therefore, P and Q are distinct points.
Examples
In architecture, when designing a bridge or a tunnel, engineers often need to divide a segment into specific ratios to determine support points or ventilation shafts. If they mistakenly calculate the partition points from opposite directions, it can lead to structural imbalances. For instance, consider a tunnel from city A to city B. If a ventilation shaft is planned to be 1/4 of the way from A to B, it will be at a different location than 1/4 of the way from B to A, which could affect air flow and safety.
