Question 204902
u = 3i - j - 2k
v = 2i + j + k


size of u (|u|) = {{{sqrt(3^2+(-1)^2+(-2)^2) = sqrt(9+1+4)=sqrt(14)}}}
size of v (|v|) = {{{sqrt(2^2+1^2+1^2) = sqrt(4+1+1)=sqrt(6)}}}

a) cosine of the angle between u and v:
cos(angle) = (dot product between u and v)/(|u|*|v|)
={{{(3*2+(-1)*1+(-2)*1)/(sqrt(14)*sqrt(6))}}}
={{{(6-1-2)/sqrt(84)}}}
={{{3/sqrt(4*21)}}}
={{{3/(2*sqrt(21))}}}
it can be rationalized so it becomes:
{{{(3/(2*sqrt(21)))*(sqrt(21)/sqrt(21))}}}
={{{3*sqrt(21)/(2*21)}}}
={{{sqrt(21)/(2*7)}}}
={{{sqrt(21)/14}}}


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