Question

In: Computer Science

Show how Rotation Matrix R transforms vectors to vectors

Show how Rotation Matrix R transforms vectors to vectors

Solutions

Expert Solution

-Rotating vector about any of the axis is vector rotation.

- Here the rotation matrix R is rotated from vector v1 to vector V2 by an angle f.

From the geometric relationship, we know that

Since

-------------(1)

Where r=length of vector and a=angle v1 within the X axis. Expanding equation (1) to 3 Dimension

From above equations, we can conclude that


Related Solutions

Show that R-1(a)R(a) = I, where I is the identity matrix and R(a) is the rotation matrix.
Show that R-1(a)R(a) = I, where I is the identity matrix and R(a) is the rotation matrix. This equation shows that the inverse coordinate transformation returns you to the original coordinate system.  
By computing both sides, show that for an m × n matrix A, vectors u and...
By computing both sides, show that for an m × n matrix A, vectors u and v ∈ Rn , and a scalar s ∈ R, we have (a) A(sv) = s(Av); (b) A(u + v) = Au + Av; (c) A(0) = 0. Here 0 denotes the zero vector. Is the meaning of 0 on the two sides identical? Why or why not? Hint: Let x = (x1, . . . , xn) and y = (y1, . ....
Consider the 90 degrees rotation matrix R = [0 −1 1 0] a) Are the eigenvalues...
Consider the 90 degrees rotation matrix R = [0 −1 1 0] a) Are the eigenvalues real? b) Are the eigenvectors real? c) Find the determinant of R. d) Find the trace of R.
Show that R-1(a) = R(-a). This equation shows that a rotation through a negative angle is equivalent to an inverse transformation.
Show that R-1(a) = R(-a). This equation shows that a rotation through a negative angle is equivalent to an inverse transformation.  
Please draw by hand the Lewis model and show how an economy transforms structurally
Please draw by hand the Lewis model and show how an economy transforms structurally
Given the matrix A (2x2 matrix taking the first two column vectors from the input file),...
Given the matrix A (2x2 matrix taking the first two column vectors from the input file), compute the followings. Λ: the diagonal matrix whose diagonal elements are the corresponding eignevalues Λii = λi for i=1,2 R: the 2x2 matrix whose ith column vector is the eigenvector corresponding to λi for i=1,2 RΛRT: the matrix compositions 1/0: the comparison between A and RΛRT (is A= RΛRT?) Your output should have seven lines where the first two lines correspond to the 2x2...
Write a program which reads the matrix A (2x2 matrix taking the first two column vectors...
Write a program which reads the matrix A (2x2 matrix taking the first two column vectors from the input file) and the vector b (the third column vector) and solve for x in Ax=b for general A. If there is a unique solution, your output should be a 2x2 matrix with two lines and two numbers per line. The output should contain numbers with up to four significant digits. If the system is unsolvable or inconsistent, then your output should...
Let A be an m x n matrix and b and x be vectors such that...
Let A be an m x n matrix and b and x be vectors such that Ab=x. a) What vector space is x in? b) What vector space is b in? c) Show that x is a linear combination of the columns of A. d) Let x' be a linear combination of the columns of A. Show that there is a vector b' so that Ab' = x'.
Consider the linear transformation T which transforms vectors x C) in the y-axis. a) Express the...
Consider the linear transformation T which transforms vectors x C) in the y-axis. a) Express the vector X = T(x), the result of the linear transformation T on x in terms of the components x and y of X. by reflection [10 Marks] b) Find a matrix T such that T(x) = TX, using matrix multiplication. Calculate the matrix product T2 represent? Explain geometrically (or logically) why it should be this. c) T T . What linear transformation does this...
Let A be a rotation matrix on a plane, please use the concept of linear transformation...
Let A be a rotation matrix on a plane, please use the concept of linear transformation to explain it doesn't have real eigenvector.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT