Vector Cross Product Calculator

The vector cross product, also known as the vector product or cross product, is a binary operation on two vectors in three-dimensional space. It has the effect of producing a vector that is perpendicular to both of the vectors being multiplied together and thus normal to the plane containing them.

Historical Background

The concept of the vector cross product was introduced as part of vector calculus in the 19th century. It is a crucial tool in physics and engineering for describing rotational effects, magnetic and electric fields, and the orientation of three-dimensional objects.

Calculation Formula

The cross product of two vectors \( \mathbf{A} = a_1\mathbf{i} + b_1\mathbf{j} + c_1\mathbf{k} \) and \( \mathbf{B} = a_2\mathbf{i} + b_2\mathbf{j} + c_2\mathbf{k} \) is given by:

\[ \mathbf{A} \times \mathbf{B} = (b_1c_2 - c_1b_2)\mathbf{i} + (c_1a_2 - a_1c_2)\mathbf{j} + (a_1b_2 - b_1a_2)\mathbf{k} \]

Example Calculation

For vectors \( \mathbf{A} = 4\mathbf{i} + 1\mathbf{j} + 3\mathbf{k} \) and \( \mathbf{B} = 4\mathbf{i} + 2\mathbf{j} + 1\mathbf{k} \), the cross product is:

\[ \mathbf{A} \times \mathbf{B} = (1 \times 1 - 3 \times 2)\mathbf{i} + (3 \times 4 - 4 \times 1)\mathbf{j} + (4 \times 2 - 1 \times 4)\mathbf{k} = -5\mathbf{i} + 8\mathbf{j} + 4\mathbf{k} \]

Importance and Usage Scenarios

The vector cross product is widely used in physics and engineering to determine the torque of a force, the magnetic force on a charged particle, and for many other applications where determining the perpendicular vector to a plane defined by two vectors is necessary.

Common FAQs

1. What does the cross product tell us?

   • The cross product provides information about the vector perpendicular to the plane formed by the two vectors. Its magnitude is proportional to the area of the parallelogram that the vectors span.

2. Is the cross product commutative?

   • No, the cross product is not commutative. \( \mathbf{A} \times \mathbf{B} \) is not equal to \( \mathbf{B} \times \mathbf{A} \); in fact, \( \mathbf{A} \times \mathbf{B} = -(\mathbf{B} \times \mathbf{A}) \).