Why sin θ is Multiplied in the Cross Product of Vectors
The sine of the angle theta; appears in the formula for the cross product of two vectors because it relates to the geometric interpretation of the cross product in three-dimensional space. This article explains the importance of sin theta in the cross product formula through an in-depth geometric and mathematical perspective.
Definition of the Cross Product
The cross product of two vectors vec{A} and vec{B} is defined as:
(vec{A} times vec{B} |vec{A}| |vec{B}| sin theta ,hat{n})
Where:
|vec{A}| and |vec{B}| are the magnitudes of vectors vec{A} and vec{B}, respectively. theta; is the angle between the two vectors. hat{n} is the unit vector perpendicular to the plane formed by vec{A} and vec{B} following the right-hand rule.Geometric Interpretation
The sin theta term in the cross product formula is crucial for understanding the resulting vector's magnitude and direction.
Magnitude of the Result
The term |vec{A}| |vec{B}| sin theta represents the magnitude of the resulting vector from the cross product. This magnitude can be understood as the area of the parallelogram formed by the two vectors.
When the angle theta; is 0^circ or 180^circ, vectors are parallel or anti-parallel, sin theta 0, and the area and thus the magnitude of the cross product is zero. When theta 90^circ, vectors are perpendicular, sin 90^circ 1, and the maximum area is obtained.Direction of the Result
The unit vector hat{n} indicates the direction of the resultant vector. The right-hand rule helps establish this direction based on the orientation of the two vectors.
Summary
In summary, the sine of the angle theta; is included in the cross product formula to ensure that the magnitude of the cross product reflects the geometric relationship between the two vectors, specifically the area of the parallelogram they define and the angle between them. This makes the cross product a vector that is both magnitude and directionally meaningful in three-dimensional space.
Understanding the Cross Product with Example
Consider two vectors vec{A} and vec{B}.
Drag them together until their tails coincide. Let this point of intersection be the origin of our 3D coordinate axes. Any two vectors vec{A} and vec{B} will always enclose a parallelogram between them, except if they both lie along a straight line, if the angle between them is either 0^circ or 180^circ. The vector cross product of vec{A} and vec{B} is a third vector. The magnitude of this third vector is equal to the area of the parallelogram enclosed between the vectors vec{A} and vec{B}. The direction of this third vector is in 3D space, towards which the face of the parallelogram is looking.The area of any parallelogram is equal to its base multiplied by its height, which gives us:
(AB sin theta)
This formula is essential for understanding the sine of the angle theta in the context of the cross product.
The direction of the cross product vector is another story. The direction at which the face of the parallelogram is looking can be found by erecting a straight line perpendicular to the face colored blue in the sketch below. However, the parallelogram has two sides. This ambiguity was resolved by defining the cross-product vector as being non-commutative—that is, vec{A} times vec{B} is not the same as vec{B} times vec{A}. Their resultant vectors point in opposite directions, even as their magnitudes are the same.
The correct direction of vec{A} times vec{B} is given by a right-hand screw rule. You can read up about it; it is all over the net.