2.9
Vector Product (Cross Product)
Vector multiplication of two vectors yields a vector product, with the magnitude equal to the product of the individual vectors multiplied by the sine…
The cross or vector product of two vectors is the product of their magnitudes and the sine of the angle. It is directed perpendicular to the vectors' plane.
Geometrically, it is the area of the parallelogram spanned by the vectors.
Using the right-hand rule, if the index finger and the middle fingers of the right-hand point the first and the second vectors, respectively, the thumb finger points to their cross product.
The cross product of the second vector with the first has the opposite direction. Hence, order matters for cross products.
The measure of a door's rotation results from a torque which is the cross product of the radial distance between the hinge and the point of application of force with the applied force.
The door does not rotate if a force acts along or opposite the direction of radial distance.
For the unit vector of an axis, its cross product with itself is zero. But with another, is the third one in cyclic-order.
The components of the cross product of vectors A and B are AyBz minus AzBy, AzBx minus AxBz, and AxBy minus AyBx.
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Q1: What is the cross product and how is it calculated geometrically?
The cross product of two vectors equals the product of their magnitudes and the sine of the angle between them, directed perpendicular to their plane. Geometrically, it represents the area of the parallelogram spanned by the two vectors. The magnitude depends on both vector sizes and their angular relationship, while direction follows the right-hand rule.
Q2: How does the right-hand rule determine cross product direction?
Point your right hand's index and middle fingers along the first and second vectors respectively. Your thumb then points in the cross product direction. If you reverse the vector order, the thumb points opposite, showing that cross products are anti-commutative. This rule resolves the ambiguity of two perpendicular directions.
Q3: Why does the order of vectors matter in cross product calculations?
The cross product is anti-commutative: reversing vector order reverses the result's direction. The cross product of vector B with vector A points opposite to the cross product of A with B. This directional reversal means A × B ≠ B × A, making order critical for correct physical interpretation.
Q4: What are the component formulas for calculating a cross product?
For vectors A and B, the cross product components are: (AyBz − AzBy), (AzBx − AxBz), and (AxBy − AyBx). These formulas use vector components in the Cartesian coordinate system to compute each component of the resulting cross product vector systematically.
Q5: When is the cross product of two vectors equal to zero?
The cross product equals zero when vectors are parallel or antiparallel, including when a vector is crossed with itself. Since the sine of 0° and 180° is zero, any vector pair aligned in the same or opposite direction produces zero cross product, regardless of magnitude.
Q6: How does torque relate to the cross product in physics?
Torque is the cross product of the radial distance vector from a hinge with the applied force vector. A door rotates only when force acts perpendicular to the radial distance; force along or opposite the radial direction produces zero torque. This demonstrates how cross product direction determines rotational effect.
Q7: What is the cross product of unit vectors in cyclic order?
Unit vectors of perpendicular axes follow a cyclic pattern: each unit vector crossed with itself yields zero, but crossed with another unit vector yields the third one in cyclic order. This property simplifies calculations and reflects the fundamental orthogonal structure of three-dimensional space.