By Kardi Teknomo, PhD .
LinearAlgebra

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Cosine Angle between Two Vectors

Cosine angle between two vectors is equal to their dot product divided by the product of their norms Vector Cos Angle .

Example:
Suppose we have Vector Cos Angle and Vector Cos Angle , the vector inner product is
Vector Cos Angle
Their norms are Vector Cos Angle and Vector Cos Angle .
Therefore, the cosine angle is Vector Cos Angle and the angle is Vector Cos Angle .

The interactive program below produces cosine angle between two vectors of the same dimension and also specify the angle in both radians and degrees. Random Example button will generate random vectors at the right format.

vector x vector y

Properties

Some important properties of related to cosine angle are

  • Two vectors are orthogonal if their dot product is zero Vector Cos Angle , which means the cosine angle is zero Vector Cos Angle
  • Two vectors are parallel if the absolute value of their dot product is equal to the product of their norms Vector Cos Angle , which means the absolute cosine angle is one Vector Cos Angle .
  • Two vectors are in the same direction if their dot product is equal to the product of their norms Vector Cos Angle , which mean the cosine angle is exactly one Vector Cos Angle .
  • Since the value of Cosine is between -1 and +1, the absolute value of vector dot product is always less than or equal to the product of their norms Vector Cos Angle . This is called Cauchy-Schwartz inequality.

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