91Ó°ÊÓ

The cross product is a fundamental operation in vector calculus that produces a vector that is perpendicular to the plane formed by the two input vectors. If you have two vectors \( \mathbf{u} = u_1 \mathbf{i} + u_2 \mathbf{j} + u_3 \mathbf{k} \) and \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k} \), their cross product \( \mathbf{u} \times \mathbf{v} \) can be calculated using the determinant of a 3x3 matrix:

• The first row contains the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \).
• The second row contains the components of \( \mathbf{u} \).
• The third row contains the components of \( \mathbf{v} \).

The result of this operation \( \mathbf{u} \times \mathbf{v} = (u_2v_3 - u_3v_2) \mathbf{i} + (u_3v_1 - u_1v_3) \mathbf{j} + (u_1v_2 - u_2v_1) \mathbf{k} \) is another vector.
Keep in mind that cross product is only defined in three-dimensional space and is specifically used in scenarios involving moments, rotational representation, and other three-dimensional concepts.
If one of the vectors is zero, as in the given exercise \( \mathbf{a} \times \mathbf{0} \), the cross product will also be the zero vector.

The curl of a vector field is an important concept in vector calculus, providing a measure of the rotation of the field at a given point. For a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), the curl is denoted by \( abla \times \mathbf{F} \) and is calculated as:

• The first component: \( \frac{ \partial R}{\partial y} - \frac{ \partial Q}{\partial z} \).
• The second component: \( \frac{ \partial P}{\partial z} - \frac{ \partial R}{\partial x} \).
• The third component: \( \frac{ \partial Q}{\partial x} - \frac{ \partial P}{\partial y} \).

The result is a new vector field.
In this exercise, we need to find the curl of the position vector \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \). The components of \( \mathbf{r} \) are such that their partial derivatives cancel each other out:

• First component: \( \frac{ \partial z}{\partial y} - \frac{ \partial y}{\partial z} = 0 \).
• Second component: \( \frac{ \partial x}{\partial z} - \frac{ \partial z}{\partial x} = 0 \).
• Third component: \( \frac{ \partial y}{\partial x} - \frac{ \partial x}{\partial y} = 0 \).

Hence, the curl \( abla \times \mathbf{r} \) is the zero vector \( \mathbf{0} \), indicating no local rotation, which is key in verifying the given identity.

Verifying a vector identity involves proving the equality of both sides of the equation using known properties and operations of vectors. In our exercise, we have the identity: \( \mathbf{a} \times (abla \times \mathbf{r}) = \mathbf{0} \).
Since we already determined that \( abla \times \mathbf{r} = \mathbf{0} \), the problem simplifies to evaluating the cross product of vector \( \mathbf{a} \) with the zero vector. This is straightforward as:

• The crossing of any vector with the zero vector results in the zero vector: \( \mathbf{a} \times \mathbf{0} = \mathbf{0} \).

In vector calculus, this specific identity also highlights something important. If the curl of a vector field is zero, it depicts a conservative field or one that has no rotational components at any point.
This conclusion confirms that the statement given is indeed true, illustrating the nondirectional curl in the context of a position vector, and concluding our verification process.