How to Find the Magnitude of a Vector from Two Points in Coordinate Geometry

In mathematics, particularly in coordinate geometry, finding the magnitude of a vector defined by two points is a fundamental skill. This process involves several straightforward steps that can help you determine the length of the vector. Let's explore the method in detail with the help of an example.

Identifying the Points

Begin by identifying the two points in the coordinate system. Let's denote these points as (A(x_1, y_1)) and (B(x_2, y_2)). For instance, consider points (A(1, 2)) and (B(4, 6)).

Determining the Vector

The vector (vec{AB}) from point (A) to point (B) can be expressed in terms of its components by subtracting the coordinates of point (A) from the coordinates of point (B).

[ vec{AB} (x_2 - x_1, y_2 - y_1) ]

For our example points (A(1, 2)) and (B(4, 6)), the vector components are calculated as follows:

[ vec{AB} (4 - 1, 6 - 2) (3, 4) ]

Calculating the Magnitude

The magnitude of the vector (vec{AB}) is calculated using the distance formula, which is derived from the Pythagorean theorem. The magnitude (or length) of the vector is given by:

[ |vec{AB}| sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2} ]

Applying the values for our example:

[ |vec{AB}| sqrt{(3)^2 (4)^2} sqrt{9 16} sqrt{25} 5 ]

This shows that the magnitude of the vector (vec{AB}) is 5.

Example: A Step-by-Step Walkthrough

Let's walk through another example to further illustrate the process. Consider points (A(3, 7)) and (B(8, 14)).

Step 1: Identify the Points

[A(3, 7), ; B(8, 14)]

Step 2: Determine the Vector

[ vec{AB} (8 - 3, 14 - 7) (5, 7) ]

Step 3: Calculate the Magnitude

[ |vec{AB}| sqrt{(5)^2 (7)^2} sqrt{25 49} sqrt{74} approx 8.6 ]

Thus, the magnitude of the vector (vec{AB}) is approximately 8.6.

Conclusion

The concept of finding the magnitude of a vector is crucial in various fields, including physics, engineering, and computer graphics. By understanding the steps involved and practicing with different examples, you can master this skill quickly. Remember, the magnitude of a vector in coordinate geometry is essentially the length of the line segment connecting two points, which can be calculated using the distance formula based on the Pythagorean theorem.

Keywords: vector magnitude, coordinate geometry, vector calculation