Find the Euclidean distance between u and v and the cosine of the angle between those vectors. State whether that angle is acute, obtuse, or 90^(circ). u = (-1, -1, 8, 0), v = (5,6,1,4)

Brennan Flores 2020-10-21 Answered
Find the Euclidean distance between u and v and the cosine of the angle between those vectors. State whether that angle is acute, obtuse, or 90. u = (-1, -1, 8, 0), v = (5,6,1,4)
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Expert Answer

yunitsiL
Answered 2020-10-22 Author has 108 answers
The Eunclidean distance between u and v is the Euclidean norm of the vector u - v. Thus, we must find
u-v=(-1,-1,8,0)-(5,6,1,4)=(-6,-7,7,-4)
and
d(u,v)=||uv||=(6)2+(7)2+72+(4)2=150=56
Furthermore, the angle 0 between these vectors is given by
cos0=u,v||uv||,
where u,v is a scalar product of u and v. So we compute
u,v=56+8+0=3
||u||=(1)2+(1)2+82+02=66
||v||=52+62+12+42=78
Therefore,
cos0=36678
which means that
0=arccos(3)66781.613
So, this angle is obtuse.
Result
Distance:
d(u,v)=56
Angle:
0=arccos(3)66781.613
It is obtuse.
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