Cynthia Vaskis
SLM521 Spring
2004
Dropin #2 Assignment
File: matrixop.htm
Lesson Title:
Math Calculations to Rotate an Object in 3D Space – Matrix Operations
When multiplying a matrix
times a vector use the following format.
The matrix has rows and
columns. Each element (or coefficient
value) in the matrix A is called a(i,j)
and has two indices each, a row number “i” and a
column number “j”. For example, the
element defined as a(1,1) means the first row and
first column element of matrix A.
Example: The element b(2,3) means the second row and third
column element of matrix B as below.
To multiply a vector P = (x,
y, z) by a matrix A perform the following
multiplications and additions:
where P’ = (x’, y’, z’) is the new vector. The multiplication of matrix A times vector P results in vector P’.
Originally, the matrix A was
formed from three simultaneous equations
a(1,1)*x +a(1,2)*y + a(1,3)*z = x’
a(2,1)*x + a(2,2)*y + a(2,3)*z = y’
a(3,1)*x + a(3,2)*y + a(3,3)*z = z’
and the coefficients (or a(i,j)
numbers) in front of the x, y, and z unknowns were placed into a matrix A and
the unknowns x, y, and z were put into a vector P.
In our case, the vector P
happens to be the 3D vector in real space.
Since we know P, the multiplication of matrix A times vector P can be
performed and the result is the new vector P’ (P prime) in 3D space.
To multiply matrix A times
matrix B, [ M(A) * M(B) = M(C)], follow the pattern of
element multiplications and additions as below.
The elements of Matrix C are
as follows and how they were obtained:
c(1,1) = a(1,1)*b(1,1)
+ a(1,2)*b(2,1) + a(1,3)*b(3,1) [first row of M(A) times first column of M(B)]
c(1,2) = a(1,1)*b(1,2) + a(1,2)*b(2,2) + a(1,3)*b(3,2) [first row of M(A) times second column of M(B)]
c(1,3) = a(1,1)*b(1,3) + a(1,2)*b(2,3) + a(1,3)*b(3,3) [first row of M(A) times third column of M(B)]
c(2,1) = a(2,1)*b(1,1) + a(2,2)*b(2,1) + a(2,3)*b(3,1) [second row of M(A) times first column of M(B)]
c(2,2) = a(2,1)*b(1,2) + a(2,2)*b(2,2) + a(2,3)*b(3,2) [second row of M(A) times second column of M(B)]
c(2,3) = a(2,1)*b(1,3) + a(2,2)*b(2,3) + a(2,3)*b(3,3) [second row of M(A) times third column of
M(B)]
c(3,1) = a(3,1)*b(1,1) + a(3,2)*b(2,1) + a(3,3)*b(3,1) [third row of M(A) times first column of M(B)]
c(3,2) = a(3,1)*b(1,2) + a(3,2)*b(2,2) + a(3,3)*b(3,2) [third row of M(A) times second column of M(B)]
c(3,3) = a(3,1)*b(1,3) + a(3,2)*b(2,3) + a(3,3)*b(3,3) [third row of M(A) times third column of
M(B)]