91Ó°ÊÓ

The cross product is a fundamental operation in vector mathematics, especially important in three-dimensional spaces. Given two vectors, such as \( \mathbf{a} \) and \( \mathbf{b} \), their cross product, denoted \( \mathbf{a} \times \mathbf{b} \), is a vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). It points in the direction given by the right-hand rule.

The magnitude of the cross product corresponds to the area of the parallelogram formed by the two vectors. Its formula in terms of components is derived from the determinant of a matrix:

• \( \mathbf{a} = (a_1, a_2, a_3) \)
• \( \mathbf{b} = (b_1, b_2, b_3) \)

The resulting vector \( \mathbf{a} \times \mathbf{b} \) is calculated as:\[\mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)\]
In the context of the original exercise, the cross product \( \mathbf{v} \times \mathbf{w} \) results in the vector \( (1, 1, -1) \). This vector is perpendicular to both \( \mathbf{v} \) and \( \mathbf{w} \).

The dot product, another essential vector operation, provides a scalar value from two vectors and measures how closely they align. It is represented as \( \mathbf{a} \cdot \mathbf{b} \) and calculated using the component-wise multiplication:

• \( \mathbf{a} = (a_1, a_2, a_3) \)
• \( \mathbf{b} = (b_1, b_2, b_3) \)

The formula is given by:\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\]
The dot product tells us about the angle between the two vectors. A dot product of zero suggests the vectors are perpendicular. For normalized vectors, it directly provides the cosine of the angle between them.

In the previous exercise, the dot product \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \) equals 2. This not only completes the triple scalar product but indicates that the vector \( \mathbf{u} \) projects onto \( \mathbf{v} \times \mathbf{w} \), showing their interaction in space.

The volume of a parallelepiped, a three-dimensional figure with parallelogram faces, can be found using vectors that define its edges. If vectors \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \) define the adjacent edges, then the volume is the absolute value of their triple scalar product.

For the specific calculation, \[V = |\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})|\]
gives the volume. It uses both the cross and dot products. First, the cross product \( \mathbf{v} \times \mathbf{w} \) finds the vector perpendicular to \( \mathbf{v} \) and \( \mathbf{w} \), representing the base area. Then, the dot product with \( \mathbf{u} \) defines the height against this base.

In this exercise, the triple scalar product yields 2. Hence, the volume of the parallelepiped is \( |2| = 2 \), reflecting how these vectors' geometric arrangement forms a solid.

Vectors in 3-dimensional space are powerful tools in representing direction and magnitude. They are composed of three components, typically expressed as \( (x, y, z) \), corresponding to the dimensions.

Working with 3D vectors involves operations such as addition, scalar multiplication, cross product, and dot product. These operations enable the analysis of spatial relationships, very useful in physics, engineering, and computer graphics.

• **Addition:** Combines vectors' respective components.
• **Scalar Multiplication:** Scales a vector by multiplying each component.
• **Cross Product:** Yields a vector perpendicular to the initial pair.
• **Dot Product:** Provides a scalar value indicating alignment.

Vectors describe not only static elements but also dynamic entities like velocity, force, and acceleration in three-dimensional environments. Understanding their manipulation is crucial for translating mathematical operations into practical applications, such as determining an object's spatial orientation and calculating the area or volume as in the exercise.