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To understand the vector cross product, think of it as a way to combine two vectors in three-dimensional space to create a new vector, which is perpendicular to both of the original vectors. This operation is essential for determining the area of parallelograms and the orientation of planes.

• The cross product of two vectors \(\mathbf{v}\) and \(\mathbf{w}\) is denoted \(\mathbf{v} \times \mathbf{w}\).
• The magnitude of \(\mathbf{v} \times \mathbf{w}\) is given by \(|\mathbf{v}| |\mathbf{w}| \sin(\theta)\), where \(\theta\) is the angle between \(\mathbf{v}\) and \(\mathbf{w}\).
• The direction of the resulting vector follows the right-hand rule.

In the original exercise, we calculate \(\mathbf{v} \times \mathbf{w}\) using determinants. This method provides a structured way by setting up a 3x3 matrix with unit vectors \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) along with the components from the vectors \(\mathbf{v}\) and \(\mathbf{w}\). Solving this determinant gives us the new vector \(\langle -3, 3, 12 \rangle\), which is perpendicular to both \(\mathbf{v}\) and \(\mathbf{w}\).

The dot product is another fundamental operation involving vectors. Unlike the cross product, the dot product results in a scalar quantity, which is why it's often called the scalar product. It measures how much one vector goes in the direction of another.

• The dot product of two vectors \(\mathbf{a} = \langle a_1, a_2, a_3 \rangle\) and \(\mathbf{b} = \langle b_1, b_2, b_3 \rangle\) is given by the formula \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\).
• It represents the product of the magnitudes of the vectors and the cosine of the angle \(\theta\) between them: \(|\mathbf{a}| |\mathbf{b}| \cos(\theta)\).
• Two vectors are orthogonal if their dot product is zero.

In our solution, we applied the dot product operation to the vector \(\mathbf{u}\) and the result of the cross product \(\langle -3, 3, 12 \rangle\). The computation \( (2 \times -3) + (1 \times 3) + (0 \times 12) \) gives a scalar result of \(-3\).

Determinants are a mathematical tool used to solve systems of linear equations, compute volumes, and find cross products among others. They transform matrices into a single scalar value that provides a lot of information about the matrix. In the context of vectors, the determinant helps in finding the cross product by organizing it in a structured format.

• A 2x2 determinant is quite simple. Given a matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant is \(ad - bc\).
• A 3x3 determinant, such as the one used in the exercise to compute the cross product, expands to a series of 2x2 determinants.

Here's the setup: you put the components of vector \(\mathbf{v}\) in the middle row and those of \(\mathbf{w}\) in the bottom row of a 3x3 matrix. The top row features the unit vectors \(\mathbf{i}, \mathbf{j}, \mathbf{k}\). This arrangement allows us to calculate the determinant using minor matrices and the concept of cofactor expansion to arrive at our desired vector \(\langle -3, 3, 12 \rangle\). This determinant calculation is crucial because it systematically arrives at the vector resulting from the cross product.