When two vectors are perpendicular, the resultant of the two vectors is the hypotenuse of a right triangle formed by the two vectors. This is a direct application of the Pythagorean theorem.
Here's a breakdown:
- Visualizing the Problem: Imagine two vectors, one pointing horizontally and the other vertically. These vectors form the two shorter sides (legs) of a right triangle.
- Pythagorean Theorem: The Pythagorean theorem states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
- Resultant Vector: The resultant vector is the diagonal of the right triangle, which is the hypotenuse.
- Magnitude: The magnitude of the resultant vector can be calculated using the Pythagorean theorem.
Example:
Let's say vector A has a magnitude of 3 units and vector B has a magnitude of 4 units, and they are perpendicular.
- Resultant Vector: The resultant vector (R) is the hypotenuse of the right triangle formed by A and B.
- Magnitude of R: Using the Pythagorean theorem, R² = A² + B² = 3² + 4² = 9 + 16 = 25. Therefore, R = √25 = 5 units.
In Conclusion: When two vectors are perpendicular, the resultant vector is the hypotenuse of the right triangle formed by the two vectors. Its magnitude can be calculated using the Pythagorean theorem.
