What is meant by antiparallel vectors? Please explain in detail, with examples.?

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Two vectors are said to be antiparallel, if they point in exactly opposite direction. The magnitudes of the two vectors NEED NOT BE THE SAME. What is important are the directions of the two vectors. If (l,m,n) are the direction cosines of a vector, its antiparallel vector will have its direction cosines as ( -l,-m,-n)

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Antiparallel vectors. In a Euclidean space, two directed line segments, often called vectors in applied mathematics, are antiparallel, if they are supported by parallel lines and have opposite directions. In that case, one of the associated Euclidean vectors is the product of the other by a negative number.

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jiss vectore mein theta 180 ka bne wo antiparallel vector hote h

n these vector are in opposite direction on same axis

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Antiparallel Vectors:

Antiparallel vectors are a pair of vectors that are exactly opposite in direction but have the same magnitude. In other words, they are parallel but point in opposite directions. This means that one vector can be obtained by multiplying the other vector by -1. Antiparallel vectors are commonly encountered in physics and mathematics, and they have several important properties.

Properties of Antiparallel Vectors:

1. Opposite Direction: Antiparallel vectors point in opposite directions. If vector A is antiparallel to vector B, then vector A points in the opposite direction of vector B.

2. Same Magnitude: Antiparallel vectors have the same magnitude. The magnitude of a vector refers to its length or size, and in the case of antiparallel vectors, they are equal.

3. Scalar Multiplication: One vector can be obtained by multiplying the other vector by -1. If vector A is antiparallel to vector B, then vector A = -1 * vector B.

4. Dot Product: The dot product of antiparallel vectors is equal to the negative of the product of their magnitudes. If vector A and vector B are antiparallel, then the dot product of A and B is equal to -|A| * |B|.

Examples of Antiparallel Vectors:

1. Force and Tension: In physics, when an object is being pulled in opposite directions by two forces of equal magnitude, the forces are antiparallel. For example, if an object is being pulled to the left with a force of 10 N and to the right with a force of 10 N, the forces are antiparallel.

2. Magnetic Field: In electromagnetism, the magnetic field created by a current-carrying wire forms loops around the wire. The direction of the magnetic field inside the loop is opposite to the direction outside the loop, making them antiparallel.

3. Electric Current: In a circuit, the direction of the current flow is often represented by a vector. If the current flows in opposite directions in two parallel wires, the current vectors are antiparallel.

4. Velocity and Displacement: In kinematics, the velocity vector and displacement vector can be antiparallel if an object is moving in the opposite direction of its displacement. For example, if an object moves 5 meters to the right and then returns 5 meters to the left, the displacement vector is antiparallel to the velocity vector.

In summary, antiparallel vectors are a pair of vectors that are opposite in direction but have the same magnitude. They can be found in various physical phenomena and have important properties such as opposite direction, same magnitude, scalar multiplication, and dot product relationships.

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