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Understanding the right-hand rule is key when dealing with the cross product of vectors. Imagine forming your right hand into a gun shape with your thumb, index finger, and middle finger all perpendicular to one another.
- **Index Finger:** Point this finger in the direction of the first vector, let's say vector \( \vec{\mathbf{A}} \). In this problem, that’s along the negative \( x \)-axis.
- **Middle Finger:** This finger should point in the direction of the second vector, such as vector \( \vec{\mathbf{B}} \). Here, it’s the positive \( z \)-axis.
- **Thumb:** Your thumb will now point in the direction of the resultant cross product vector. For \( \vec{\mathbf{A}} \times \vec{\mathbf{B}} \), the thumb shows the positive \( y \)-axis direction.
Now, let's reverse it. Point your index finger along \( \vec{\mathbf{B}} \) and your middle finger along \( \vec{\mathbf{A}} \). Your thumb will point to the negative \( y \)-axis, giving you the direction for \( \vec{\mathbf{B}} \times \vec{\mathbf{A}} \). Remember, swapping the order of vectors in a cross product flips the vector direction!

Vector direction in 3D space is crucial in vector operations. With vectors \( \vec{\mathbf{A}} \) and \( \vec{\mathbf{B}} \), if you change their order, the direction of their cross product changes. For \( \vec{\mathbf{A}} \times \vec{\mathbf{B}} \), the vector points differently from \( \vec{\mathbf{B}} \times \vec{\mathbf{A}} \).
- When you cross \( \vec{\mathbf{A}} \) and \( \vec{\mathbf{B}} \), remember they follow the right-hand rule to determine the direction.
- Direction helps in locating objects or forces in physics problems like when dealing with torque or magnetic forces.
Hence, knowing the direction of vectors means understanding instinctively how the vector behaves in physical space. The correct calculation ensures accurate representation and solution in diverse applications.

To compute the magnitude of a cross product of two vectors, such as \( \vec{\mathbf{A}} \) and \( \vec{\mathbf{B}} \), use the formula: \[ |\vec{\mathbf{A}} \times \vec{\mathbf{B}}| = |\vec{\mathbf{A}}| |\vec{\mathbf{B}}| \sin(\theta) \] where \( \theta \) is the angle between the two vectors.
- Given the vectors \( \vec{\mathbf{A}} \) is along the negative \( x \)-axis and \( \vec{\mathbf{B}} \) is on the positive \( z \)-axis, they are perpendicular, making \( \theta = 90^\circ \).
- The sine of \( 90^\circ \) is 1, simplifying the magnitude to just the product of the magnitudes of \( \vec{\mathbf{A}} \) and \( \vec{\mathbf{B}} \).
Thus, whether it's \( \vec{\mathbf{A}} \times \vec{\mathbf{B}} \) or \( \vec{\mathbf{B}} \times \vec{\mathbf{A}} \), their magnitudes are identical. Only their directions differ. This property of cross products is crucial across fields—from physics to engineering—ensuring vector forces and directions are properly calculated.