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The cross product is a mathematical operation that occurs between two vectors. It results in a third vector, which is perpendicular to the plane formed by the original vectors. To compute the cross product of two vectors, each represented as three components in space, we typically use a determinant formula. This involves organizing the components into a 3x3 matrix and solving for the resulting vector.

A unique property of the cross product is that it tells us about the rotational effect and direction of the perpendicular vector. The vector \(\mathbf{u} \times \mathbf{v}\) results from the vectors \(\mathbf{u} = \langle u_1, u_2, 0 \rangle\) and \(\mathbf{v} = \langle 0, 5, 0 \rangle\). Thus, the cross product simplifies to \(\langle 0, 0, 5u_1 \rangle\).

• The resulting vector is perpendicular to the vectors \(\mathbf{u}\) and \(\mathbf{v}\).
• This reflects the fact that the direction of the result is dependent on the initial vectors' directions.

The magnitude of a vector, also often called the vector's length, gives an idea of its size irrespective of its direction. To find the magnitude of a given vector, we sum the squares of its components and then take the square root of the result.

In our exercise, after calculating the cross product \(\mathbf{u} \times \mathbf{v}\ = \langle 0, 0, 5u_1 \rangle \), we find its magnitude by simply taking the absolute value of the z-component, which is \(\|5u_1\|\). Since the vector \(\mathbf{u}\) can vary as it rotates in the xy-plane, it can take values of \(-3 \le u_1 \le 3\). This variation gives us insight into the possible sizes of the cross product vector's magnitude.

The maximum magnitude occurs when \(u_1 \) is at its extremities of either 3 or -3, providing a max magnitude of 15. Meanwhile, the minimum value will be zero when \(u_1 \) is 0.

Understanding the direction of a vector helps visualize where it points in space. With cross products, this direction is crucial because it shows the vector normal (perpendicular) to the plane of the initial vectors.

In this exercise, the direction of \(\mathbf{u} \times \mathbf{v}\) is affected by \(u_1\), the x-component of vector \(\mathbf{u}\). Since the resulting vector is \(\langle 0, 0, 5u_1 \rangle\), it indicates that the direction lies entirely along the z-axis. Whether it points in the positive or negative direction depends on the sign of \(u_1\). This results in:

• A positive z-direction, if \(u_1\) is positive.
• A negative z-direction, if \(u_1\) is negative.

The z-axis direction forms a right-hand rule with the vectors \(\mathbf{u} \) and \(\mathbf{v}\) helping find the precise orientation.

Rotation in the xy-plane involves vectors moving in circles centered around the z-axis. This type of vector movement is essential in various physics and engineering applications.

Here, vector \(\mathbf{u}\) describes such a rotation. It maintains a constant length but changes direction as it sweeps through different angles. This is expressed as its components, \(u_1\) and \(u_2\), satisfy the circle equation \(u_1^2 + u_2^2 = 9\).

Rotation in the xy-plane implies:

• Vector \(\mathbf{u}\) remains within the plane while changing its orientation.
• The length, or magnitude, stays constant.
• Different orientations of \(\mathbf{u}\) affect the cross-product results.

This setting establishes a dynamic system where the orientation of \(\mathbf{u}\) continuously evolves, influencing the cross product's magnitude and direction.