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3D Maths - Matrix Algebra
Sometimes, there arises a question with developers what if we are interested only in direction, not location or length. The scenario results in usage of only direction which results in vector formation.
A vector may be represented with list of elements in vertical and horizontal format called column matrix or row matrix. In a column matrix, an ordered list of numbers is written in a column, whereas in row matrix the elements are written in row format.
The dimension of matrix is defined with number of rows and columns combined together. The representation of matrix is mentioned below −
$$\begin{bmatrix}\:\:\:2\:.\:9 \\-4\:.\:6 \\\:\:\:0\:.\:0 \end{bmatrix}$$The numbers represented in matrix in row or column format is called as an element. The numbers are actually real numbers. The number of elements defined in matrix as a vector is called as dimension.
A row matrix is defined as an ordered list of numbers written in a row. Here is the demonstration of a row matrix which is mentioned below −
$$(12.5,\:-9.34)$$Vectors in 3D algebra is always represented with column matrices. Now let us demonstrate that with the following example −
$$Column\:Matrix\begin{bmatrix}\:\:\:2\:.\:9 \\-4\:.\:6 \\\:\:\:0\:.\:0 \end{bmatrix}\begin{bmatrix}-13.45 \\\:92.4\\\:\:\:2.0\\-42.01 \end{bmatrix}\begin{bmatrix}2\\0\\5\\1 \end{bmatrix}$$The dimension of first matrix is 3*2, the second matrix is 4*1 and third matrix is also same with a value of 4.
The elements of a column matrix can be variables. The following schematic representation shows variable matrix −
$$\begin{bmatrix}u\\v\\w\end{bmatrix}\begin{bmatrix}x_0\\x_1\\x_2\end{bmatrix}\begin{bmatrix}z_1\\z_2\\z_3\end{bmatrix}$$The first element in a column matrix is usually represented with index “0” and sometimes “1”.
Type of Matrices
The following are the types of matrices −
Row Matrix
A row matrix is a matrix which defines only one row. Its order can be considered as 1 X C, where C is the number of columns. The demonstration of row matrix is mentioned below with a dimension of 1 X 5 −
$$[3\:5\:7\:9\:11]$$Column Matrix
A column matrix is a matrix with a dimension of only one column. It is represented by an order of R X 1, where R is represented with number of rows. The demonstration of column matrix of the order 3 X 1 −
$$\begin{bmatrix}2\\5\\7 \end{bmatrix}\\$$Square Matrix
The square matrix is the number of rows which is equal to the number of columns is referred as a square matrix. Here's a square matrix of the order 2 X 2 −
$$\begin{bmatrix}4\:\:\:9\\15\:\:\:2\\ \end{bmatrix}$$Diagonal Matrix
A diagonal matrix is defined as a square matrix where all the elements are 0 except for those in the diagonal from the top left corner to the bottom right corner. The demonstration of diagonal matrix of order 4 X 4 −
$$\begin{bmatrix}8\:\:0\:\:0\:\:0\\0\:\:5\:\:0\:\:0\\ 0\:\:0\:\:2\:\:0\\ 0\:\:0\:\:0\:\:9\end{bmatrix}$$
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