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The cross product is a mathematical operation applied to two vectors that results in another vector. This resulting vector is perpendicular to the plane defined by the original vectors. The formula for the cross product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is:

• \( \mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2) \mathbf{i} + (a_3b_1 - a_1b_3) \mathbf{j} + (a_1b_2 - a_2b_1) \mathbf{k} \)

Let's consider the exercise, where we compute \( \mathbf{u} \times \mathbf{v} \), where \( \mathbf{u} = 2\mathbf{i} \) and \( \mathbf{v} = 2\mathbf{j} \). Substitute these values into the cross product formula. We get:

• \( \mathbf{u} \times \mathbf{v} = 4(\mathbf{k}) \)

This simplifies because the unit vector cross product \( \mathbf{i} \times \mathbf{j} \) results in \( \mathbf{k} \). The cross product gives us important orientation information about the vectors, especially in geometry and physics.

The dot product is an operation that takes two vectors and returns a scalar. It measures how much one vector extends in the direction of another. The dot product for vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \) is:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \)

In the exercise, we find \( (\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = 8 \). The process involved taking the cross product result \( 4\mathbf{k} \) and \( \mathbf{w} = 2\mathbf{k} \), then performing the dot product:

• \( 4\mathbf{k} \cdot 2\mathbf{k} = 8 \)

This operation illustrates how aligned the cross product vector is with \( \mathbf{w} \). The dot product is zero if the vectors are perpendicular and is maximized if they point in the same direction.

The volume of a parallelepiped defined by vectors \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \) can be found using the scalar triple product. The scalar triple product involves the cross product and the dot product. The formula is

• \( \text{Volume} = (\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} \)

In the exercise, the volume of the parallelepiped was calculated by first finding \( \mathbf{u} \times \mathbf{v} \) and then taking the dot product with \( \mathbf{w} \), resulting in 8. This volume represents the capacity of the parallelepiped whose edges are defined by the vectors \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \). The volume is always a positive value and reflects the three-dimensional space occupied by the solid.

Unit vectors are vectors with a magnitude of 1 that point in a specific direction. They are typically denoted by \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \), representing the x, y, and z axes respectively.

• \( \mathbf{i} = (1, 0, 0) \)
• \( \mathbf{j} = (0, 1, 0) \)
• \( \mathbf{k} = (0, 0, 1) \)

In the context of the exercise, our vectors \( \mathbf{u} = 2\mathbf{i} \), \( \mathbf{v} = 2\mathbf{j} \), and \( \mathbf{w} = 2\mathbf{k} \) are multiples of these unit vectors. Despite being scaled, these vectors still preserve the direction set by the unit vectors. Unit vectors are foundational in vector operations, helping define direction while ignoring scale. They are useful in various applications ranging from physics to computer graphics.