Find a transformation matrix to rotate a vector in R2 through an angle of 30 degrees. Then use it to rotate the vector (3, 7).

Find all vectors (x,y,z) that are orthogonal to the vectors (1,2,3) and (4,5,6). Find the vector in two different ways

Let the basis B = {b1, b2} and basis C = {c1, c2} where vectors b1 = (7 -2). b2 = 2 -1 , c1 = 4 1  c2= 5 2 Find the change of coordinates matrix from B to C. (b) Find the change of coordinates matrix from C to B.

Explain the Combination of basis vectors.

Calculate the dot product of u = (-4, -2, 7, -2) and v = (-3, 1,5,8). u. V =

Calculate div(F) and curl(F). F = (yz, 6xz, 7xy) (Give an exact answer. Use symbolic notation and fractions where needed.) div(F) = = (Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.) curl(F): =

Consider the following vectors: u = (1, −2, −1), v = (-2, 1, 8) Without using the cross product, find a vector n which is orthogonal to both u and U. You need to use a notion from chapter 5 to set up a system of linear equations, then use RREF to solve the system. Explain why this method works.