Cross Product of Two Vectors

Find the cross product of the vectors $\mathbf{A} = \langle 1, 3, 4 \rangle$ and $\mathbf{B} = \langle 2, 7, -5 \rangle$.

The cross product is a binary operation on two vectors in three-dimensional space, resulting in another vector that is orthogonal, or perpendicular, to the plane containing the original vectors. When calculating the cross product of two vectors, it's important to remember that the resulting vector's direction is determined by the right-hand rule. This means that if you point your index finger in the direction of the first vector and your middle finger in the direction of the second vector, your thumb will indicate the direction of the resulting cross product vector.

The magnitude of the cross product vector represents the area of the parallelogram formed by the original vectors. Understanding how to compute the cross product is essential in physics and engineering, especially in contexts where rotational forces and torques are involved. In addition, cross products feature prominently in computer graphics and in the analysis of vector fields, where they can be used to compute the curl of a vector field.

In this problem, calculating the cross product involves using the determinant of a special 3x3 matrix that includes the components of the vectors in question. Mastering this determinant approach will not only aid in solving similar problems but also deepen your comprehension of vector multiplication in general.

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