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The cross product is a binary operation on two vectors in three-dimensional space. The result of this operation is another vector that is perpendicular to the plane formed by the original vectors. Given vectors \( \mathbf{a} \) and \( \mathbf{b} \), the cross product \( \mathbf{a} \times \mathbf{b} \) is calculated using the determinant of a 3x3 matrix:

• The top row is filled with the unit vectors \( \hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}} \).
• The second row contains the components of \( \mathbf{a} \).
• The third row contains the components of \( \mathbf{b} \).

The resulting vector from the cross product operation thus embodies:

• A direction defined by the right-hand rule, meaning if the fingers of your right-hand point from \( \mathbf{a} \) to \( \mathbf{b} \), your thumb points in the direction of \( \mathbf{a} \times \mathbf{b} \).
• A magnitude equal to the area of the parallelogram spanned by \( \mathbf{a} \) and \( \mathbf{b} \).

The key aspect of the cross product is its ability to produce this perpendicular vector, which has applications in finding torque, rotation, and in our exercise, it's used within nested cross product expressions to contribute to a dot product calculation.

The dot product, also known as the scalar product, is a way to multiply two vectors to result in a scalar. This operation is expressed as:\( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta) \)Where:

• \( |\mathbf{a}| \) and \( |\mathbf{b}| \) are the magnitudes (or lengths) of the vectors.
• \( \theta \) is the angle between the vectors.
• \( \mathbf{a} \cdot \mathbf{b} \) provides a scalar that reflects how much of one vector goes in the direction of another.

Alternatively, in terms of components, the dot product can be calculated by multiplying corresponding components of the vectors and summing them:\( \mathbf{a} \cdot \mathbf{b} = a_xb_x + a_yb_y + a_zb_z \)In vector algebra, this operation is particularly helpful for determining projections, testing orthogonality (if \( \mathbf{a} \cdot \mathbf{b} = 0 \), the vectors are perpendicular), and as seen in the exercise, it acts as the final calculation step to simplify nested vector operations.

Vector operations are foundational in vector algebra and involve actions like addition, scalar multiplication, cross product, and dot product. These operations expand our ability to manipulate and understand vector quantities in space. When adding vectors like \( 2\mathbf{a} + \mathbf{b} \), you're essentially combining corresponding components to produce a new vector. For instance, if \( \mathbf{a} = a_x \hat{\mathbf{i}} + a_y \hat{\mathbf{j}} + a_z \hat{\mathbf{k}} \) and \( \mathbf{b} = b_x \hat{\mathbf{i}} + b_y \hat{\mathbf{j}} + b_z \hat{\mathbf{k}} \), then \( 2\mathbf{a} + \mathbf{b} \) results in:\[(2a_x + b_x) \hat{\mathbf{i}} + (2a_y + b_y) \hat{\mathbf{j}} + (2a_z + b_z) \hat{\mathbf{k}}\]Scalar multiplication involves multiplying each component of a vector by a scalar, scaling its magnitude without changing the direction unless the scalar is negative, which reverses it.These combinations of vector operations empower the expression of complex concepts succinctly. In exercises like the given one, understanding how these operations interlink—such as using cross products to yield vectors that are themselves operands in further vector operations—allows one to manage intricate vector algebra challenges efficiently.