Answer
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"Box products" refer to the cross product or vector product of two vectors. The cross product is a mathematical operation that produces a new vector that is perpendicular to both of the original vectors.
The significance of the cross product lies in its ability to provide information about the orientation, direction, and magnitude of vectors in three-dimensional space. It is particularly useful in solving problems involving torque, angular momentum, and determining the normal vector to a plane.
The cross product of two vectors, denoted as A × B, is evaluated using the following formula:
A × B = |A| |B| sin(θ) n
where |A| and |B| are the magnitudes of the vectors A and B, θ is the angle between the vectors, and n is a unit vector perpendicular to the plane formed by A and B, following the right-hand rule.
For example, if we have two vectors A = (2, 3, 4) and B = (5, -1, 2), we can calculate their cross product as follows:
A × B = |A| |B| sin(θ) n
= |A| |B| sin(θ) (n1, n2, n3)
By substituting the values, we can calculate the cross product:
A × B = (2, 3, 4) × (5, -1, 2)
= (3*2 - 4*(-1), 4*5 - 2*2, 2*(-1) - 3*5)
= (10, 16, -17)
So, the cross product of A and B is (10, 16, -17). This new vector is perpendicular to both A and B and provides information about their relative orientation and direction in three-dimensional space.
