1) For which values of k are the following vectors u and v orthogonal?
a) u = (2,1,3) , v = (1,7,k)
b) u = (k,k,1) , v = (k,5,6)
2) Let u,v be orthogonal unit vectors. Prove that d(u,v) = 2^(1/2)
(The questions are unrelated)

Two vectors are given by A = -3i + 7j -4k and
B = 6i -10j + 9k. Evaluate the quantities:
a) cos^-1 [A.B/AB] ( the . between the A.B is a dot product)
b) sin^-1 [ |A x B|/AB]
c) Which gives the angle between the vectors.
I haven't a clue as to what to do with this problem. Do I use the cross products of unit vectors t

In the following case, state if the following set of vectors are linearly independent or linearly dependent. Justify your answer.
G = {[(1, -1), (-1, 0)],[(1, -4), (1, 0)],[(1, -6), (1, 0)],[(0, 0), (1, 0)]}
These are four 2x2 matrices

Please see the attached file.
22. For the case of plane polar coordinate r, theta, write the unit vectors and e_theta in terms of i and j. Hence show that and. By starting with r = re_r, and differentiating, rederive the expressions for the components of the velocity and acceleration vectors.

Two vectors having equal magnitudes (A) makes an angle z with each other. Find the magnitude and direction of the resultant and prove that the resultant of two equal vectors bisects the angle between them.

In C[-pi, pi] with inner product defined by (6), show that cos mx and sin nx are orthogonal and that both are unit vectors. Determine the distance between the two vectors.
(6) (f,g) = (1/pi)* the integral from -pi to +pi of f(x)g(x)dx
This is all from Linear Algebra With Applications by Steven J. Leon, Sixth Edition. Than

1 Given a = 9i - 5j and b = 7i-4j, express i and j in terms of a and b
2 Given a=<4,5,-3> and b =<4,-2,2> determine whether a and b are parrallel, perpendicular, or neither.
3 Given F = 4i -2k;..... P(0,1,0) and Q(4,0,1) find the work W done by the force (F)moving a particle in a straight line from P to Q.
4 Given a

Suppose {v_1, v_2, v_3} is linearly independent set of vectors in R^n. Determine which of the following sets of vectors are linearly independent and which are linearly dependent. If a set is linearly dependent give, a linear dependence relation. (Use the following technique. If {w_1, w_2, w_3} denotes one of the sets below, solv