91Ó°ÊÓ

The cross product is a mathematical operation that takes two vectors and produces another vector that is perpendicular to both. For two vectors \( \mathbf{u} = u_1\mathbf{i} + u_2\mathbf{j} + u_3\mathbf{k} \) and \( \mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k} \), the cross product \( \mathbf{u} \times \mathbf{v} \) is determined using the determinant:\[\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \end{vmatrix}\]Hence, we perform operations to find the vector perpendicular to both input vectors. A significant aspect of the cross product is that it not only gives a vector but also a sense of orientation (right-hand rule). Besides, the magnitude of the cross product represents the area of the parallelogram spanned by the original two vectors.

The dot product is an operation on two vectors that results in a scalar. For vectors \( \mathbf{u} = u_1\mathbf{i} + u_2\mathbf{j} + u_3\mathbf{k} \) and \( \mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k} \), the dot product is calculated as:\[\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3\]This computation results in a single number, representing how much one vector extends in the direction of the other. When the dot product is zero, the vectors are orthogonal. In this context, the dot product is used in calculating the scalar triple product, which helps determine coplanarity and volume of structures like parallelepipeds.

Coplanarity of vectors involves determining if multiple vectors lie within the same plane. For three vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \), they are said to be coplanar if their scalar triple product is zero:\[ \mathbf{a} \cdot ( \mathbf{b} \times \mathbf{c}) = 0\]Being coplanar implies that there is no volume created by the vectors, as they do not form a three-dimensional space. It's a critical evaluation when working with 3D vector problems to understand if the vectors lie flat or extend into a space. For example, three coplanar vectors won't form a parallelepiped.

A parallelepiped is a 3D shape formed by three vectors, and its volume can be found using the scalar triple product:\[V =| \mathbf{a} \cdot ( \mathbf{b} \times \mathbf{c}) |\]This formula calculates the magnitude of the product, which gives the volume of the parallelepiped spanned by vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \). If the result is zero, it confirms the vectors are coplanar, meaning no volume exists. This is a practical way to determine spatial arrangements with vectors and is fundamental in physics and engineering when assessing structures and forces.