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The cross product is an operation between two vectors in three-dimensional space. It results in a third vector that is perpendicular to both original vectors. This new vector is useful in finding the area of shapes as well as in physical applications like torque. To calculate the cross product of two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), we use the formula:

• \( \mathbf{a} \times \mathbf{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle \)

By following these steps, you effectively ensure that the result is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \).
This is clearly demonstrated in the problem where the vectors \( \mathbf{u} = \langle 1,1,1 \rangle \) and \( \mathbf{w} = \langle 4,2,7 \rangle \) are used to calculate \( \mathbf{u} \times \mathbf{w} = \langle 5, -3, -2 \rangle \).

The dot product is a way to multiply two vectors that results in a scalar. This operation helps us understand the relationship between the angles of two vectors. If the dot product is zero, the vectors are perpendicular. For two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), the dot product is calculated as:
\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]This computation gives us a single scalar value. It's used in physics to imply work done.
In our original problem, the dot product was used twice: once to find \( \mathbf{v} \cdot (\mathbf{u} \times \mathbf{w}) = -1 \) and again for \( \mathbf{w} \cdot (\mathbf{u} \times \mathbf{v}) = 1 \). It shows the scalar result when a cross product vector is multiplied with another vector.

Vectors are foundational elements in mathematics and physics, representing magnitude and direction. Unlike scalars, vectors are essential in defining quantities that have both size and orientation.

• Basic Format: Written as \( \langle x, y, z \rangle \), pointing in three-dimensional space.
• Properties: Directions, like force, velocity, or any other directional component.

Operations on vectors, like addition, subtraction, dot products, and cross products, reveal more complex relationships between quantities.
In the original problem, the vectors \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \) serve as examples of how vectors can form the building blocks for calculating volumes and understanding spatial relationships.

A parallelepiped is a six-faced figure (also known as a polyhedron) where each face is a parallelogram. The shape formed in the triple scalar product question is a parallelepiped.
This geometric figure has applications in understanding three-dimensional space and volumes within these spaces. In terms of vectors, a set of three non-coplanar vectors can define a parallelepiped.

• The intersection point of vectors represents the vertex of the parallelepiped.
• The vectors act like the edges emanating from one vertex.

Calculating the volume of a parallelepiped involves using a scalar triple product of these vectors, as seen in the problem involving \( \mathbf{v}, \mathbf{u}, \) and \( \mathbf{w} \).

Calculating volume with vectors involves understanding the alignment and orientation within three-dimensional space. The scalar triple product \( \mathbf{a} \cdot ( \mathbf{b} \times \mathbf{c} ) \) is crucial for finding the volume of a parallelepiped spanned by three vectors.
This operation results in a scalar, representing the volume. If the computed volume is zero, the vectors are coplanar, indicating no volume—as they lie on a plane.
The initial problem shows how this is done by computing the volume defined by the vectors \( \mathbf{v}, \mathbf{u}, \) and \( \mathbf{w} \), where calculated results were both \(-1\) and \(1\), indicating the magnitude of space they span and the possibility of orientation flipping.