Adding and Subtracting Vectors: Why We Use Them to Find Displacement and Not the Pythagorean Theorem

When discussing motion, particularly in physics and engineering, displacement is a crucial concept that we often need to calculate. Displacement, as the change in location from one point to another, is not just about distance but includes direction as well. Therefore, it is best measured as a vector, which is a quantity with both magnitude and direction. In this article, we will explore why we add and subtract vectors to find displacement, and why the Pythagorean theorem doesn't suffice for all scenarios.

Understanding Displacement and Vectors

Displacement is change in location. If an object moves from point A to point B, the displacement vector can be written as AB. If there is an additional displacement from point B to point C, the resultant displacement is AC. The vector addition follows the triangle law, which implies that the segments AB and BC can be added to form the resultant vector AC.

Triangle Law of Vector Addition

According to the triangle law of vector addition, the sum of two vectors can be obtained by placing the tail of the second vector at the head of the first vector. This forms a triangle with the resultant vector as the closing side. This method allows us to find the resultant displacement even when the vectors are not perpendicular to each other. The magnitude of the resultant vector can be determined using the law of cosines when the angle between the vectors is not 90 degrees.

Pythagorean Theorem and Vector Addition

The Pythagorean theorem, which is widely known in geometry, is a fundamental principle stating that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is typically written as c2 a2 b2.

However, the Pythagorean theorem works only if the vectors are perpendicular to each other. If two vectors are not perpendicular, the Pythagorean theorem cannot be applied to find the magnitude of the resultant vector. For vectors that are not perpendicular, we must use other methods such as the law of cosines, which is defined by the formula c2 a2 b2 - 2ab cos(C).

Visualizing the Pythagorean Theorem

To illustrate how the Pythagorean theorem applies to perpendicular vectors, consider the top half of the diagram below. Here, vector AC is the resultant vector that can be found using c2 a2 b2. However, if the vectors are not perpendicular, as shown in the bottom half of the diagram, the Pythagorean theorem would not be applicable.

Conclusion

Vector addition is essential for calculating displacement because it accounts for both magnitude and direction. In contrast, the Pythagorean theorem is limited to right-angled triangles and perpendicular vectors. For non-perpendicular vectors, we must use more sophisticated methods such as the law of cosines. Understanding these principles is crucial for analyzing motion and solving related problems in physics and engineering.

Best of luck with your studies!