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The cross product is a mathematical operation used in vector algebra to find a vector that is perpendicular to two given vectors. It is especially useful in three-dimensional space. Given two vectors \( \mathbf{a} \) and \( \mathbf{b} \), the cross product \( \mathbf{a} \times \mathbf{b} \) results in a new vector that is orthogonal to both \( \mathbf{a} \) and \( \mathbf{b} \).

The cross product is calculated using the determinant of a matrix formed by the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \), followed by the components of the two original vectors. For example, here the cross product \( \mathbf{w} \times \mathbf{a} \) is calculated as follows:

• Organize the vectors into a 3x3 matrix with the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) on the top row.
• Align the components of \( \mathbf{w} \) and \( \mathbf{a} \) in the subsequent rows.
• The resulting calculation of the determinant gives \( \langle 0, 5, -2 \rangle \).

Therefore, using the cross product helps us find vectors like \( \mathbf{u} \) that are orthogonally related to the given vector.

The dot product, also known as the scalar product, is a fundamental operation in vector algebra that provides a scalar value from two vectors. For two vectors \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \), the dot product is calculated as:

• \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \)

This result can determine the angle between the vectors or check their orthogonality.

A key point in this exercise was verifying the orthogonality of vectors \( \mathbf{u} \) and \( \mathbf{v} \) with \( \mathbf{w} \):

• If \( \mathbf{u} \cdot \mathbf{w} = 0 \) and \( \mathbf{v} \cdot \mathbf{w} = 0 \), then \( \mathbf{u} \) and \( \mathbf{v} \) are perpendicular to \( \mathbf{w} \).
• The zero value in dot products confirms no projection of one vector onto another, solidifying the concept of orthogonality.

This simple multiplication followed by addition allows us to easily test for perpendicularity.

Orthogonal vectors are vectors that are perpendicular to each other. This means the angle between them is \( 90^\circ \) or \( \frac{\pi}{2} \) radians. When two vectors are orthogonal, the dot product between them is zero. This is a crucial property used in various mathematical and physical applications.

In the presented problem, the task was to find vectors \( \mathbf{u} \) and \( \mathbf{v} \) such that they are not only perpendicular to the given vector \( \mathbf{w} \) but also to each other. Here’s how this was achieved:

• First, find one vector, \( \mathbf{u} \), orthogonal to \( \mathbf{w} \) using the cross product.
• Then calculate another cross product with \( \mathbf{w} \) and \( \mathbf{u} \) to get \( \mathbf{v} \), ensuring \( \mathbf{v} \) is also orthogonal to both \( \mathbf{w} \) and \( \mathbf{u} \).
• The dot product conservation is used to confirm the orthogonality: \( \mathbf{u} \cdot \mathbf{v} = 0 \).

This structured method of checking ensures all vectors meet the condition of orthogonality and further strengthens understanding of their properties.