Understanding Perpendicular Vectors
To find a vector that is perpendicular to a given vector represented as ai + bj + ck, we need to follow some fundamental principles of vector mathematics.
1. Definition of Perpendicular Vectors
- Two vectors are perpendicular if their dot product is zero.
- If we have a vector V = ai + bj + ck, we want to find another vector U = xi + yj + zk such that V · U = 0.
2. Dot Product Calculation
- The dot product is calculated as:
- V · U = ax + by + cz.
- Set this equal to zero to find the condition for perpendicularity:
- ax + by + cz = 0.
3. Choosing a Suitable Vector
- To find a perpendicular vector, you can choose arbitrary values for two of the components (x, y, z) and solve for the third.
- For example, if you choose x = 1 and y = 0, you can solve for z:
- a(1) + b(0) + cz = 0 → z = -a/c (if c ≠ 0).
4. Practical Example
- Given V = 3i + 4j + 5k:
- Let’s choose x = 4 and y = -3.
- To find z, solve: 3(4) + 4(-3) + 5z = 0 → z = -12/5 = -2.4.
- Thus, U = 4i - 3j - 2.4k is perpendicular to V.
5. Conclusion
- By utilizing the properties of dot products, you can efficiently determine a vector that is perpendicular to any given vector in 3D space.

so some more information is required to find the required vector