Become a Linear Algebra Expert

back
Dot Product

Dot product is very difficult to describe with words exactly what it is. However, it will help if you understand that the dot product between Force and Displacement is Work.
Dot product has two definitions. They both yield the same result, but each of them are used in different situations.

0.0.1. Unit Vector With the knowledge of norm of a vector, it is possible to find the unit vector. Unit vector is a vector that has the same direction, but has magnitude of 1.
Unit vector is calculated by dividing the vector by the norm. It is indicated with a hat on top of the vector.

\begin{equation}
\uvec{x} = \frac{\vec{x}}{||\vec{x}||}
\end{equation}

  1. $\vec{x}\cdot\vec{y} = ||\vec{x}|| ||\vec{y}|| cos\theta$
  2. $\vec{x}\cdot\vec{y} = x_1y_1 + x_2y_2 + … +x_ny_n$

The first equation is generally used more for physics problem, where the angle between the vectors, and magnitudes of the vectors are usually given. The second definition is more used for mathematics, where the components of the vectors is usually given.

With the second expression for dot product, the length of a vector can be expressed as
\begin{equation}
||\vec{x}|| = \sqrt{\vec{x}\cdot\vec{x}}
\end{equation}

Share with: