Determining the Direction and Distance of Point C Relative to Point A

Today, we will explore a classic problem in coordinate geometry that involves vector analysis. Specifically, we are given two points, A and B, and another point, C, with relative positions defined in terms of their distances from B. By using vector algebra, we will determine the direction and distance of C with respect to A. This step-by-step guide will be suitable for anyone interested in understanding vectors and their applications.

Given Data and Initial Setup

Let's start with the given data:

A is 40 meters to the south-west of B. C is 40 meters to the south-east of B.

To simplify the problem, we will place point B at the origin of the coordinate system. Let's denote the unit vectors along the East and North directions as i and j, respectively. Based on the given data, we can represent the positions of A and C relative to B using vectors:

AB 40cos 45° i - 40sin 45° j

BC 40cos 45° i 40sin 45° j

Calculation of Vector AC

Next, we need to find the resulting vector AC. This can be done by adding the vectors AB and BC:

AC AB BC

AC (40cos 45° i - 40sin 45° j) (40cos 45° i 40sin 45° j)

AC 80cos 45° i

AC 80/√2 i

Interpretation of the Resulting Vector

The simplified form of AC indicates that C is located 80/√2 meters to the east of A. Simplifying further:

80/√2 40√2 ≈ 56.57 meters

This means that C is approximately 56.57 meters to the east of A.

Alternative Interpretation Using Coordinate System

Alternatively, you can think of the problem geometrically. If B is placed at the origin of an X-Y chart, with the positive X-axis pointing towards the East and the positive Y-axis pointing towards the North, then:

A is on the line OB making an angle of 45° with the negative X-axis. C is on the line OC making an angle of 45° below the positive X-axis.

The vector AC, which is parallel to the X-axis towards the negative X-axis direction, indicates that C is to the east of A. The distance |AC| is 80 meters, which simplifies to 40√2 meters.

Concluding Statement

Based on the analysis, we can conclude that C is 40√2 meters, or approximately 56.57 meters, to the east of A. This problem not only demonstrates the power of vector analysis but also highlights the importance of coordinate systems in solving geometric problems.

Thank you for your interest in this topic. Feel free to ask any further questions or seek more detailed explanations.