Cross Product of Vectors

Find the cross product of vectors $\mathbf{a}$ and $\mathbf{b}$, where $\mathbf{a} = 3\mathbf{i} + 5\mathbf{j} - 7\mathbf{k}$ and $\mathbf{b} = 2\mathbf{i} - 6\mathbf{j} + 4\mathbf{k}$.

The cross product is a fundamental operation in vector calculus that takes two vectors and returns a vector that is perpendicular to both. It is particularly useful in physics, computer graphics, and engineering for finding a perpendicular direction or calculating torque. The cross product of two vectors also provides a way to determine the area of the parallelogram that the vectors span. Understanding the cross product involves both geometric insight and algebraic manipulation.

In terms of calculation, the cross product requires determinant evaluation, which can often be remembered by using the right-hand rule and the structure of a 3x3 matrix. The vectors' components are used to fill this matrix alongside the unit vectors for the coordinate directions. By using the determinant, one can calculate the orthogonal vector result of the cross product.

Understanding the properties of the cross product is crucial, such as its anti-commutative nature (switching the order of the vectors changes the sign of the result) and how it relates to vector magnitudes and angles. It's also essential to visualize how cross products can indicate rotational effects or orientations in physical systems.

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