Understanding Perpendicular Vectors
In vector mathematics, two vectors are said to be perpendicular if their dot product equals zero. For the vector given, 4i - 3j, we will find a vector that fulfills this condition.
Given Vector
- The vector provided is: 4i - 3j
- This can be represented as: (4, -3) in Cartesian coordinates.
Finding a Perpendicular Vector
To find a vector that is perpendicular to (4, -3), we can determine a vector (x, y) such that:
- The dot product (4 * x) + (-3 * y) = 0
Solution Steps
- Rearranging the equation gives: 4x = 3y
- To find a simple solution, we can let x = 3 and y = 4.
Resulting Perpendicular Vector
- Therefore, one perpendicular vector is: 3i + 4j
General Form of Perpendicular Vectors
- Any scalar multiple of a perpendicular vector will also be perpendicular. Thus, we can express it as:
- k(3i + 4j) where k is any real number.
Examples of Perpendicular Vectors
- If k = 1: 3i + 4j
- If k = -1: -3i - 4j
- If k = 2: 6i + 8j
Conclusion
In summary, the vector 3i + 4j is perpendicular to 4i - 3j. This method can be applied to any vector to find its perpendicular counterparts.