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The cross product, denoted as \( \mathbf{a} \times \mathbf{b} \), is a crucial operation in vector calculus when dealing with three-dimensional vectors. Its primary function is to find a vector that is perpendicular to the plane formed by the input vectors. This is particularly useful in physics and engineering to determine forces and torques.

For vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), the resultant cross product is a new vector expressed as:

• \( \mathbf{a} \times \mathbf{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle \)

This process involves setting up and solving a determinant, capturing the essence of crossing vectors in space. It's noteworthy that if \( \mathbf{a} \times \mathbf{b} = \mathbf{0} \), the vectors are parallel.

In vector mathematics, perpendicular vectors (or orthogonal vectors) have a dot product of zero. This means that they are at a perfect 90-degree angle to each other. The significance of finding a vector that is perpendicular to two other vectors lies in various applications.

Consider a scenario in computer graphics, where you need to find a normal vector to shade a surface correctly. By calculating the cross product of two known edges of a surface, you obtain a vector perpendicular to that surface, aiding in accurate rendering.

To ensure a vector is perpendicular to the given vectors \( \mathbf{a} \) and \( \mathbf{b} \):

• Compute \( \mathbf{a} \times \mathbf{b} \).
• Check if the dot product with both vectors is zero.

It's also important in electrical engineering, where perpendicular vectors can represent different phases of an AC system.

The calculation of cross products involves determinants, which are structured as a 3x3 matrix in this context. The determinant is essentially a scalar value that helps find the orientation and area (magnitude of the vector) in 3D space.
The setup for finding \( \mathbf{a} \times \mathbf{b} \) is:

• \[ \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} ewline 1 & -2 & -3 ewline -3 & 2 & 0 \end{vmatrix} \]

Here, each symbol \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) represents a unit vector in the x, y, and z directions respectively. By expanding this determinant, you calculate each of the components of the resulting vector:

• The \( \mathbf{i} \)-component uses the 2x2 minors of the remaining elements after excluding the row and column of \( \mathbf{i} \).
• The \( \mathbf{j} \)-component includes adjusting signs as per the cofactor expansion method.
• The \( \mathbf{k} \)-component is similar to the \( \mathbf{i} \)-component but uses different elements.

This systematic method ensures accuracy in finding the vector that is orthogonal to the initial pair of vectors.