SOLUTION: Hi all, Im still having trouble with these vector problems, I cant solve the problems listed below for the following: Let ~u = 3~i - ~j - 2~k and ~v = 2~i + ~j + ~k.(Again, the ~

Algebra ->  Equations -> SOLUTION: Hi all, Im still having trouble with these vector problems, I cant solve the problems listed below for the following: Let ~u = 3~i - ~j - 2~k and ~v = 2~i + ~j + ~k.(Again, the ~      Log On


   

Question 204902: Hi all, Im still having trouble with these vector problems, I cant solve the problems listed below for the following:
Let ~u = 3~i - ~j - 2~k and ~v = 2~i + ~j + ~k.(Again, the ~ symbols mean that there is and arrow symbol above the value in the positive direction)
I need to:
a) Calculate the cosine of the angle between ~u and ~v.
b) Fin the cross product of ~u and ~v.
Some help with steps and explanations on how to solve would be great.
Thanks, -Nick
Answer by alicealc(293) About Me  (Show Source):
You can put this solution on YOUR website!
u = 3i - j - 2k
v = 2i + j + k

size of u (|u|) = sqrt%283%5E2%2B%28-1%29%5E2%2B%28-2%29%5E2%29+=+sqrt%289%2B1%2B4%29=sqrt%2814%29
size of v (|v|) = sqrt%282%5E2%2B1%5E2%2B1%5E2%29+=+sqrt%284%2B1%2B1%29=sqrt%286%29
a) cosine of the angle between u and v:
cos(angle) = (dot product between u and v)/(|u|*|v|)
=%283%2A2%2B%28-1%29%2A1%2B%28-2%29%2A1%29%2F%28sqrt%2814%29%2Asqrt%286%29%29
=%286-1-2%29%2Fsqrt%2884%29
=3%2Fsqrt%284%2A21%29
=3%2F%282%2Asqrt%2821%29%29
it can be rationalized so it becomes:
%283%2F%282%2Asqrt%2821%29%29%29%2A%28sqrt%2821%29%2Fsqrt%2821%29%29
=3%2Asqrt%2821%29%2F%282%2A21%29
=sqrt%2821%29%2F%282%2A7%29
=sqrt%2821%29%2F14

b) i cross j = k, j cross k = i, k cross i = j
j cross i = -k, k cross j = -i, i cross k = -j
cross product between i and i = between j and j = between k and k = 0
so,cross product between u and v:
(3i*j) + (3i*k) + (-j*2i) + (-j*k) + (-2k*2i) + (-2k*j)
= 3k + (-3j) + 2k + (-i) +(-4j) + 2i
= i - 7j + 5k