Vector Proof – Angle Between Two Vectors

Prove that:

cos\theta =\frac { \underline { a } \cdot \underline { b }  }{ \left| \underline { a }  \right| \left| \underline { b }  \right|  }

Firstly, look at the image below:

vector image

Also know that:

cosC=\frac { { a }^{ 2 }+{ b }^{ 2 }-{ c }^{ 2 } }{ 2ab }

\left| \underline { a }  \right| \left| \underline { a }  \right| ={ \underline { a }  }^{ 2 }\\ \\ \left| \underline { b }  \right| \left| \underline { b }  \right| ={ \underline { b }  }^{ 2 }\\ \\ \left| \underline { b } -\underline { a }  \right| \left| \underline { b } -\underline { a }  \right| ={ \left( \underline { b } -\underline { a }  \right)  }^{ 2 }

From the image, you’ll be able to see that:

a=\left| \underline { b }  \right| \\ \\ b=\left| \underline { a }  \right| \\ \\ c=\left| \underline { b } -\underline { a }  \right| \\ \\ cosC=cos\theta ,\quad \therefore \quad C=\theta

Now:

vector image complete

Other posts you may be interested in...