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An antiparallel vector is the opposite of a parallel vector. Since an anti parallel vector is opposite to the vector, the dot product of one vector will be negative, and the equation of the other vector will be negative to that of the previous one. The antiparallel vectors are a subset of all parallel vectors. They are also known as antiparallel vectors, as they are always opposite to the direction of a given vector and never cross it.
Antiparallel vectors:
Two parallel vectors in the Euclidean Space pointing in opposite directions are called Antiparallel vectors. In this case, one of the vectors can be put equal to the product of the other vector with any negative number. As it is already known that for two vectors to be parallel, they need not have the same length and only need to have the same component of direction. In the same way, for two vectors to be Antiparallel, they need not have the same length; they are only required to be parallel and opposite in direction.
Derived Relations/Examples of Antiparallel vectors:
In this triangle ABC, the altitude is AD. Now, in ∆ABD, the right angle is made by BD and in ∆ADC, the right angle is made by DC. Consider these lines to be vectors instead. Then, by the triangle law of vector addition, BD and DC will have opposite directions. They also have an angle of 180° between them which is a requirement for Antiparallel vectors. Hence, BD and DC are theoretically known to be Antiparallel vectors.
This is an image of a circumcircle of a triangle ABC where DE is the tangent of the circumcircle at the vertex of the triangle. There is a theorem that DE will have a direction opposite to that of BC in the vector form. Moreover, they are both parallel vectors. Hence, DE and BC are Antiparallel vectors since they are opposite in direction. O
In the same figure, if O is the centre of the circle, then OA will be the radius of the circumcircle at the vertex and it will be perpendicular to DE. Therefore, there is a theorem in Euclidean mathematics stating that the radius of the circumcircle at the vertex will be perpendicular to all the Antiparallel vectors of BC (in the same way it is perpendicular to DE).
Difference between Antiparallel vectors and opposite vectors:
Two vectors are said to be opposite or negative vectors if they have the same magnitude but act in the opposite direction to each other. On the other hand, Antiparallel vectors need not have the same magnitude but should point in opposite directions and also have an angle of 180° between them. This means that one of the two vectors should be equal to the negative non-zero scalar multiple of the first vector.
For example: Ā and -Ā are opposite or negative vectors. However, Ā and -7Ā are antiparallel vectors and -7 is non zero as well as scalar.
A negative vector set is always an antiparallel vector set but the converse might not be true.
Difference between parallel and Antiparallel vectors:
Parallel vectors point towards the same direction whereas antiparallel vectors point in different directions. Parallel vectors have an angle of 0° in between them whereas antiparallel vectors have an angle of 180°.
Conditions for two vectors to be Antiparallel:
The conditions for a vector set to be antiparallel are:
They should point in opposite directions.
They should have an angle of 180°
The first vector should be a non-zero scalar multiple of the second.
Conclusion
Antiparallel vectors are the ones that are parallel in nature but opposite in direction. In these, one of the vectors is the non-zero scalar multiple of the other vector. The conditions for an antiparallel vector is that they point in opposite directions to one another and have an angle of 180° in between them. Any vector that has any other angle can not be an antiparallel vector. This means that the slopes of Antiparallel vectors will be the same. Antiparallel vectors can be concluded by a phrase that they are parallel and oppositely directed.
