91Ó°ÊÓ

The cross product is a fundamental operation in vector algebra. It takes two vectors and produces another vector that is perpendicular to both original vectors. Mathematically, for two vectors \(\mathbf{a}\) and \(\mathbf{b}\), their cross product \(\mathbf{a} \times \mathbf{b}\) is defined such that:

• The magnitude of the resulting vector is equal to the area of the parallelogram spanned by \(\mathbf{a}\) and \(\mathbf{b}\).
• The direction follows the right-hand rule: if you point your right-hand fingers from \(\mathbf{a}\) towards \(\mathbf{b}\), your thumb points in the direction of \(\mathbf{a} \times \mathbf{b}\).

This operation is crucial for determining the orientation and perpendicular direction in 3D space and is commonly used for physics applications like torque and rotational dynamics.
The specific cross product formula is given by:\[\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} = (a_2b_3 - a_3b_2)\hat{i} + (a_3b_1 - a_1b_3)\hat{j} + (a_1b_2 - a_2b_1)\hat{k}\]

Coplanarity refers to the condition where vectors lie on the same plane. For vectors to be coplanar, it typically means that when arranged tail-to-head, they all reside in a two-dimensional plane in three-dimensional space. This concept is significant in determining structural behavior in geometry and physics.
In vector mathematics:

• For three vectors \( \mathbf{a}, \mathbf{b}, \text{ and } \mathbf{c} \) to be coplanar, their scalar triple product should be zero. This aligns with the condition \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \).
• The scalar triple product, when equal to zero, implies that the vectors do not span a volume but rather a flat surface.

In our exercise, vectors \(\mathbf{y}\) and \(\mathbf{r}\) inherently shared common origins due to their construction, reinforcing their coplanarity along with any third vector that forms a plane with them.

The scalar triple product is a useful concept that combines both dot product and cross product operations. It is used to determine if three vectors are coplanar, as well as to calculate the volume of the parallelepiped they span.

• The scalar triple product formula for three vectors \(\mathbf{a}, \mathbf{b}, \text{ and } \mathbf{c}\) is expressed as \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \).
• Geometrically, it calculates the volume of the parallelepiped formed by the vectors: if the result is zero, the vectors are coplanar.

In our case, assessing \(\mathbf{x} \cdot (\mathbf{y} \times \mathbf{r}) = 0\) confirmed that the vectors \(\mathbf{x}, \mathbf{y}, \text{ and } \mathbf{r}\) do not form a three-dimensional structure, but rather lie on the same plane. This concept also underscores the importance of understanding the relationships between vectors in space.