BackCross Product: Computation and Properties
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Cross Product
Introduction to the Cross Product
The cross product is a mathematical operation used to multiply two vectors in three-dimensional space, resulting in a new vector that is perpendicular to both original vectors. This operation is distinct from the dot product, which yields a scalar.
Dot Product: yields a scalar.
Cross Product: yields a vector that is always perpendicular to the original vectors.
Key Property: The cross product vector is always perpendicular to both input vectors.
Definition and Formula
The cross product of vectors and is denoted as .
It is computed using the determinant of a matrix:
Where are the unit vectors in the , , and directions, respectively.
How to Calculate the Cross Product
Write the matrix of each vector's components.
Repeat the first two columns outside the matrix for easier computation.
For each component, calculate the sum of the products of the diagonals (downward) and subtract the sum of the products of the diagonals (upward).
Multiply and combine the components diagonally as shown in the determinant.
Example Calculation
Given and , compute :
Expanding the determinant:
Practice Problems
Given and , find .
Given and , compute .
Summary Table: Dot Product vs. Cross Product
Operation | Result Type | Geometric Meaning |
|---|---|---|
Dot Product () | Scalar | Projection of one vector onto another |
Cross Product () | Vector | Vector perpendicular to both and |
Additional info: The cross product is used in physics and engineering to find torque, angular momentum, and the normal vector to a plane defined by two vectors.
