Vectors in three dimensional space

The cartezian x-y plane represents the horizontal and vertical directions. The other dimension out of the board ( or out of the screen when viewed from a computer) is the z direction. Previously a column vector with two row was used and the most obvious extension two represent a three dimensional vector would be to add another row; indeed this is commonly used, but for reasons which will become clearer later, mathematicians write vectors in terms of the vectors i, j, and k, which are unit vectors in the x,y, and direction of the Cartezian coordinate system

\( \widehat{\mathbf{i}}=\begin{bmatrix} 1\\ 0\\ 0\\ \end{bmatrix} \) space \( \widehat{\mathbf{j}}=\begin{bmatrix} 0\\ 1\\ 0\\ \end{bmatrix} \) space \( \widehat{\mathbf{k}}=\begin{bmatrix} 0\\ 0\\ 1\\ \end{bmatrix} \)