91Ó°ÊÓ

To understand the cross product of vectors, we must first be familiar with the concept of vector components. A vector is a quantity that has both magnitude (length) and direction. In three-dimensional space, a vector can be represented by three components along the x, y, and z axes. These components are typically enclosed in angle brackets and written as \( \langle a, b, c \rangle \), where \( a \), \( b \), and \( c \) are the magnitudes of the vector in the x, y, and z directions, respectively.

For instance, if we have vector \( \mathbf{u} \) with components \( \langle -4, 1, 1 \rangle \), this means that \( \mathbf{u} \) has an x-component of -4, a y-component of 1, and a z-component of 1. Understanding the orientation and value of these components is crucial because they are the building blocks for vector operations, such as the cross product, which is used to find a vector that is perpendicular to both original vectors in 3D space.

The cross product of two vectors results in a third vector that is perpendicular to the plane formed by the first two vectors. The cross product formula for vectors \( \mathbf{u} \) and \( \mathbf{v} \) in three-dimensional space is given by:

\[ \mathbf{u} \times \mathbf{v} = \langle (u_y v_z - u_z v_y), -(u_x v_z - u_z v_x), (u_x v_y - u_y v_x) \rangle \]
Here, \( u_x, u_y, u_z \) are the components of vector \( \mathbf{u} \) and \( v_x, v_y, v_z \) are the components of vector \( \mathbf{v} \) along the x, y, and z axes, respectively. The cross product formula is essentially a determinant of a matrix formed by the unit vectors along the x, y, and z axes and the components of vectors \( \mathbf{u} \) and \( \mathbf{v} \). The direction of the resulting vector is determined by the right-hand rule, which is a useful mnemonic for visualizing the orientation of this new vector.

Calculating the cross product requires following the cross product formula systematically. By substituting the respective components of the vectors into the formula and evaluating the expressions, you can find the components of the new vector that is orthogonal to both initial vectors.

For example, given vectors \( \mathbf{u} = \langle -4, 1, 1 \rangle \) and \( \mathbf{v} = \langle 0, 1, -1 \rangle \), the cross product \( \mathbf{u} \times \mathbf{v} \) is calculated as follows:
\[ \mathbf{u} \times \mathbf{v} = \langle (1 \cdot -1 - 1 \cdot 1), -(-4 \cdot -1 - 1 \cdot 0), (-4 \cdot 1 - 1 \cdot 0) \rangle \] which simplifies to \( \langle -2, 4, -4 \rangle \).

In contrast, \( \mathbf{v} \times \mathbf{u} \) reverses the order of multiplication, which according to the cross product properties, results in a vector that is equal in magnitude but opposite in direction to \( \mathbf{u} \times \mathbf{v} \), hence the result \( \langle 2, -4, 4 \rangle \). This step-by-step approach helps verify that the vectors are indeed perpendicular by making use of the geometric and algebraic properties of the cross product.