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In three-dimensional space, the cross product is a binary operation on two vectors. It generates a perpendicular vector to both vectors. The two vectors are parallel if the cross product of their cross products is zero; otherwise, they are not. The condition that two vectors are parallel if and only if they are scalar multiples of one another can also be used.
Vectors with the same or opposite direction are called parallel vectors. Two vectors are said to be parallel if and only if their angle is 0 degrees. Parallel vectors are also known as collinear vectors.
Two parallel vectors will always be parallel to each other, but they can point in the same or opposite directions.
Cross Product of Two Parallel Vectors
Any two parallel vectors’ cross product is a zero vector. Consider a and b, two parallel vectors. The angle between them is then equal to θ = 0. The term “cross product” is defined as
a × b = |a| |b| sin θ n
|a| |b| sin 0 n
|a| |b| (0) n (sin 0 = 0)
=0
The magnitude of vector A multiplied by the magnitude of vector B, multiplied by the sine of the angle formed by vectors n is a unit vector perpendicular to the plane formed by vectors A and B, and directed in a direction perpendicular to the plane formed by vectors A and B.
The cross product of two vectors is zero vector perpendicular to the plane formed by the two vectors because the angle between them is zero. We cannot define a plane and take the direction of the vector obtained from cross product in accordance with a right handed screw because there is only one vector. The null vector obtained from two parallel vectors does not have a unique direction.
Cross Product of Antiparallel Vectors
Anti-parallel vectors are parallel vectors that are in the opposite direction.
Two directed line segments, also known as vectors in applied mathematics, are antiparallel in a Euclidean space if they are supported by parallel lines and have opposite directions. In this case, one of the associated Euclidean vectors is the negative product of the other.
Finding a vector w that is perpendicular to both u and v in space, given two non-parallel, nonzero vectors u and v, is very useful. A cross product operation can be used to generate such a vector. This section defines the cross product before delving into its properties and uses.
(If two vectors point in the same direction, they are parallel; if they point in opposite directions, they are anti-parallel.)
AB/(|A||B|)=0, if A is perpendicular to B, and vice versa if AB/(|A||B|)=0 if A and B are perpendicular.
Cross Product of Orthogonal Vectors
The concept of “orthogonality” is crucial. A quick scan of your current surroundings will undoubtedly reveal a plethora of perpendicular surfaces and edges (including the edges of this page). Quickly check for orthogonality with the dot product the vectors u and v are perpendicular if and only if u.v=0
Two orthogonal vectors’ dot product is zero. The two column matrices that represent them have a zero dot product. The relative orientation is all that matters. The dot product will be zero if the vectors are orthogonal.
Orthogonal vectors’ scalar product vanishes, while antiparallel vectors’ scalar product is negative. The vector product of two vectors is a vector perpendicular to both vectors. Its magnitude is calculated by multiplying the magnitudes of the two angles by the sine of the angle between them.
Because the direction perpendicular to both vectors maximizes the volume, the cross-product is roughly orthogonal to a and b.
Conclusion
We conclude in this article that, “Parallel vectors are vectors that have the same or exact opposite direction. Any two parallel vectors’ cross product is a zero vector. Simply take a common factor out of one vector and multiply it by the other vector to see if they are parallel. Another option is to see if their cross product is zero.
