Angle between Two Vectors

Angle between Two Vectors:

The angle between two vectors is the angle between their ends. The angle between two vectors can be found either by using dot product (scalar product) or cross product (vector product). The angle between two vectors always lies between 0° and 180°.

What is Angle between two vectors?

A vector quantity is a physical quantity having both magnitude and direction. When two vectors act on a particle, then the resultant action on the particle will depend on the angle between those vectors.

Some properties of vectors for angle calculation:

 The parallel vectors are the two vectors with the same direction. Anti-parallel vectors are the two vectors having opposite directions.

A vector is denoted by an arrow parallel to the direction of the vector.

Equal vectors are the two vectors having the same magnitude and direction. Negative vectors are the two vectors with the same magnitude and the opposite direction.

Dot product:

The Dot product is also known as the scalar product of vectors. The Dot product has only magnitude but no direction.

Then Dot product of A and B can be written as,

A.B=ABcos

Special Cases:

• When the angle between vectors is 0 degree. That is =θ0°

=>ABcos=θ

=>ABcos 0°

=>AB          [since cos 0° = 1}

• When the angle between vectors is 90 degree. That is =90°

=>ABcosθ

=>ABcos90°

=>AB(0)           [cos90°=0]

=>0

• When the angle between vectors is 180 degree. That is =180°

=>ABcos

=>ABcos180°

=> –AB              [since cos180° = -1]

Angle between two vectors formulas

The angle between two vectors using dot product is, 

cosθ= [(a.b)/(a .b)]

The angle between the two vectors using cross product is,

sin = [a×b /(ab)].