How to Find a Vector Perpendicular to Two Vectors Using the Cross Product?

The cross product of two vectors results in a vector that is perpendicular to both of the original vectors. Here's how to find it:

Understanding the Cross Product

The cross product is a binary operation on two vectors in three-dimensional space. It produces a vector that is orthogonal (perpendicular) to both of the original vectors. The magnitude of the resulting vector is equal to the area of the parallelogram formed by the two original vectors.

Calculating the Cross Product

1. Represent the vectors: Let's say you have two vectors:

   • a = (a₁, a₂, a₃)
   • b = (b₁, b₂, b₃)

2. Calculate the determinant: The cross product a x b is calculated using the following determinant:

   | i  j  k |
   | a1 a2 a3 |
   | b1 b2 b3 |

   • i, j, and k are the unit vectors along the x, y, and z axes, respectively.

3. Expand the determinant: Expanding the determinant, we get:

   a x b = (a₂b₃ - a₃b₂) i + (a₃b₁ - a₁b₃) j + (a₁b₂ - a₂b₁) k

4. The resulting vector: The resulting vector is perpendicular to both a and b.

Example

Let's say you have two vectors:

• a = (1, 2, 3)
• b = (4, 5, 6)

Calculating the cross product:

| i  j  k |
| 1  2  3 |
| 4  5  6 |

Expanding the determinant:

a x b = (2 6 - 3 5) i + (3 4 - 1 6) j + (1 5 - 2 4) k

a x b = -3 i + 6 j - 3 k

Therefore, the vector (-3, 6, -3) is perpendicular to both vectors (1, 2, 3) and (4, 5, 6).

Practical Insights

• The cross product is used in various fields, including physics, engineering, and computer graphics.
• It's essential for calculating torque, angular momentum, and magnetic force.
• In computer graphics, it's used to find the normal vector of a surface, which is crucial for lighting and shading calculations.